151217

Ruby


周期性をもつ差分方程式

Lyness の周期5の漸化式
Xn = (1 + Xn-1) / Xn-2
や
Todd の周期8の漸化式
Xn = (1 + Xn-1 + Xn-2) / Xn-3
が知られているが、
Xn = (1 + Xn-1 + … + Xn-k-1) / Xn-k
とするとどうなるか調べてみた。

def f(m, k)
  i = 0
  ary = []
  a = (1..k).map{|i| i.to_r}
  while i < m
    j = a.shift
    a << (a.inject(:+) + 1r) / j
    ary << a[0]
    i += 1
  end
  ary
end

(2..20).each{|k| p f(50, k)}

出力結果
[(2/1), (3/1), (2/1), (1/1), (1/1), (2/1), (3/1), (2/1), (1/1), (1/1), (2/1), (3/1), (2/1), (1/1), (1/1), (2/1), (3/1), (2/1), (1/1), (1/1), (2/1), (3/1), (2/1), (1/1), (1/1), (2/1), (3/1), (2/1), (1/1), (1/1), (2/1), (3/1), (2/1), (1/1), (1/1), (2/1), (3/1), (2/1), (1/1), (1/1), (2/1), (3/1), (2/1), (1/1), (1/1), (2/1), (3/1), (2/1), (1/1), (1/1)]
[(2/1), (3/1), (6/1), (5/1), (4/1), (5/3), (4/3), (1/1), (2/1), (3/1), (6/1), (5/1), (4/1), (5/3), (4/3), (1/1), (2/1), (3/1), (6/1), (5/1), (4/1), (5/3), (4/3), (1/1), (2/1), (3/1), (6/1), (5/1), (4/1), (5/3), (4/3), (1/1), (2/1), (3/1), (6/1), (5/1), (4/1), (5/3), (4/3), (1/1), (2/1), (3/1), (6/1), (5/1), (4/1), (5/3), (4/3), (1/1), (2/1), (3/1)]
[(2/1), (3/1), (4/1), (10/1), (9/1), (8/1), (7/1), (5/2), (37/18), (113/72), (57/56), (79/35), (526/185), (18963/4181), (67304/6441), (37562/4503), (177765/20777), (494764/79163), (5851931/2532313), (98046897/45144158), (261614425/190777398), (74481337/67916388), (1690057423/692495764), (1696378550597/623559343589), (6029565661142/1139261940115), (13712628104169/1312968619795), (621174943307012/77915302440773), (71980376442658082/7925102517572309), (1220041251046289217/226572524776683353), (6808715737286399320/3033736421891847407), (278630559500696573527/125418050577090731281), (11019507432146370995407/9223863798245458847158), (174204079984755385555873/140835077573252800580346), (18366442192414548442209701/7291487386385377749456456), (552874077649080643363207827/206415147079612715612646856), (35031590414511209797116628687/5629503952292954716675146899), (2432185804023615805583578312666/242223132110381300183165448179), (111800777414205821245942921389435/14121346654349647407737981310541), (31467586462712162138370474484598024/3347136459697471197497855062579761), (703778898330217990181025180098953342/154427996208554466480441541821046029), (21044531098292478509443342252430712933/9237271794359226423000234872368818649), (840719233011935293567051450687190733580/386154171803116477271422619215887566927), (102774074027276961211036742400189999374357/96498847037682201795763591985878555422823), (153505051230391161880444022117277167545766697/107289719999554203025706844826175619523766318), (6551158666190530259293789258635118527240565843/2630906210874893189352752426289855056847134514), (4670274020677296004976968730204360232289761203/1698658846312210834513988490328246524328954620), (1024793296700404240309241466269964459609807605927/142295113198168366911437876741346856134778695620), (44280370177323083058111703560019096793231770612066637/4713384019048658109138418101328393700395574116574189), (53208185402841696667952365361990974226230438558034239406/6512005409394100834299349070617100585185219669235785399), (31041951558547504289487999239332201730951839756840461459/3312201568295082377106282008491510404674854370037177521)]
[(2/1), (3/1), (4/1), (5/1), (15/1), (14/1), (13/1), (12/1), (11/1), (17/5), (101/35), (212/91), (67/39), (34/33), (29/11), (305/101), (21609/5353), (24232/3551), (19365/1139), (5962/493), (407863/30073), (40032259/3201219), (23678969/2877084), (16769363/6016060), (5509076/1749305), (225522763/110530873), (222443977807/161660170817), (222758941797/177082499507), (353972569208/111822367397), (1935169695505/689430561482), (100853774248107/21421069674086), (15685576217810201/1668146858901517), (259432691560573495/15478911347646841), (150120874090465188/13703206952236957), (868625797836432311/56930697912867217), (24801147593323800016/2187638189780123137), (1650734362004120825743/280633075963062985609), (66731777348512460990099/25172256322007377816735), (181322640015427341663047/54983217614871951438645), (461783233266974395426496/291531263074097863423011), (4966393354804454269651457/3905925741310579428754271), (398194484238274426532709448/238900923088369554285038401), (219703123115287524035768176239/66035215855836770262830455121), (2097649253016897543774282086705/781707561339154230112761109933), (9594532980452702089000597314615/1527594147535644205881615871504), (118476215614325953767110974358611/10071004155645849088014615561904), (6794453192625747551379716903409577/451992118641775570678966797603364), (1722202174404452221138729428624127613/155870029742938731278207837535913404), (307883385315341368517088370641861585967/18308255175573818625070518968581598817), (3243911589633974059182254961707061360341/366038326102779659249537873064130965835)]
[(2/1), (3/1), (4/1), (5/1), (6/1), (21/1), (20/1), (19/1), (18/1), (17/1), (16/1), (13/3), (113/30), (601/190), (430/171), (277/153), (141/136), (319/104), (4911/1469), (274220/67913), (734578/129215), (597699/59555), (7259/277), (5232/319), (9897979/522203), (435146276/22444907), (4288585149/265047341), (1394714030/142643449), (9823251/3150833), (9125176/2187521), (11037348003/3900565109), (2803598661805/1465986624158), (19367769632641/13739356654074), (68367123252181/46289903464458), (8500619088125/2070449734014), (4445476173929/1456227524196), (6505938068183537/1420783209883404), (4357062722866631743/533314911671863245), (38415545546363972614/2419232702733846215), (124431813615337562157/4995781842590209481), (2685911975576458516652/191489025388969414375), (12660050488810499073021/563691278508118406875), (249585179705928846763144/13221355438950108177113), (86418649445756523242652171/7267352150783297119598899), (669852722023857534790602575/114174944518871734044858419), (9318151120963631762902679902/3131226964506626785007329643), (146093045818621497020706561414/32489980102234546359332109211), (3158186133577781329643002601875/1572451205690144722947699736583), (44396647043841361969600140719532/29678477064707427742033124594837), (1627136922041558026704248019140725/1084352328742923529767976010636879)]
[(2/1), (3/1), (4/1), (5/1), (6/1), (7/1), (28/1), (27/1), (26/1), (25/1), (24/1), (23/1), (22/1), (37/7), (884/189), (109/27), (84/25), (66/25), (43/23), (263/253), (1434/407), (30541/8177), (102438/24089), (22849/4251), (20275/2574), (122057/8514), (1308700/33927), (4022029/188571), (77677948/3128271), (220558425/8276611), (1085395622/43344553), (2499134339/129713773), (7573449840/692917589), (1161700341/347252165), (5416649638/1040242955), (706681167591/191905994789), (17496656858228/6809429995275), (118663634680513/64596100617975), (1419394710600430/957145190663537), (964964001057105/551488011650293), (135361465413876/25834153340599), (312758183314675/92736223826777), (6792659013187538/1449944314499657), (1063963357669548179/141161476573659571), (460790468957642689650/33768741361571174909), (26264645198452128448827/1046191031416240369085), (60934689161564109314716/1759855266127988043425), (1348681984100949428219/78544057359952706175), (4688936173881504297262/152394319894506884775), (141518713259124534328759/5105681520430454148325)]
[(2/1), (3/1), (4/1), (5/1), (6/1), (7/1), (8/1), (36/1), (35/1), (34/1), (33/1), (32/1), (31/1), (30/1), (29/1), (25/4), (157/28), (1175/238), (4753/1122), (3697/1056), (2705/992), (355/186), (181/174), (579/145), (3268/785), (843311/184475), (30195706/5584775), (124525599/17571841), (106978208/10000385), (19098139/960275), (17544726/321275), (939774/34933), (4930061/157681), (3377266564/98426441), (3675997132415/106993155557), (503717856447862/16756409879135), (274057925977363/12615080900707), (24256523141784/2059561175761), (2104646128265/600486766783), (986284288889/157931775210), (3653084227343/798817783830), (1750041678929763/530258922731860), (4024625492365502513/1682453205673541140), (3006474016625505038885/1657779763597075379006), (12514645783961352095447/7856198506227328470466), (6892101526051222757209/3323718248677041966696), (1743648286669829289295/265816550027471657832), (120558131450428231983/32305336704985333567), (506642449752424694390/103139712575977235419), (887302178890067432377573/121632593422628847204567)]
[(2/1), (3/1), (4/1), (5/1), (6/1), (7/1), (8/1), (9/1), (45/1), (44/1), (43/1), (42/1), (41/1), (40/1), (39/1), (38/1), (37/1), (65/9), (1297/198), (5539/946), (4636/903), (3775/861), (591/164), (145/52), (478/247), (731/703), (10753/2405), (388827/84305), (35543684/7184083), (71980753/12839402), (59882361/8750450), (20581303/2231025), (6097304/428475), (13961272/519825), (39211801/524127), (260430827/7860443), (17793750715/464561859), (1481401452296/34899858633), (59541685382071/1352252187481), (198124113728064/4773396939311), (49082531414165/1431425105803), (9043853561306/382592869229), (5076732198275/409260190436), (4010554996521/1108191537188), (105768014411469/14526261397709), (10575203480835413/1926836263675381), (638118911064394707/156335259776964524), (18586034242483801423/6138928183120343476), (1049591022426036275263/459392253868133445936), (299435351427685633699/163427444503884344976)]
[(2/1), (3/1), (4/1), (5/1), (6/1), (7/1), (8/1), (9/1), (10/1), (55/1), (54/1), (53/1), (52/1), (51/1), (50/1), (49/1), (48/1), (47/1), (46/1), (41/5), (1013/135), (9712/1431), (4167/689), (1168/221), (1911/425), (92/25), (17/6), (551/282), (1123/1081), (4672/943), (210755/41533), (13178457/2459564), (19893179/3372492), (5569655/811176), (1370693/159432), (74350/6279), (50795/2737), (2321796/65569), (61682753/618773), (52496695/1311664), (2262904677/49232368), (350497121729/6857841247), (3303082390088/61067180071), (8578482044499/160810078205), (235929608704/4934897945), (4525978821/119895323), (1897330398/75532165), (126532341793/9827968985), (598280851568/161589842729), (2264131599763/272318616685)]
[(2/1), (3/1), (4/1), (5/1), (6/1), (7/1), (8/1), (9/1), (10/1), (11/1), (66/1), (65/1), (64/1), (63/1), (62/1), (61/1), (60/1), (59/1), (58/1), (57/1), (56/1), (101/11), (6052/715), (4019/520), (3515/504), (12107/1953), (10216/1891), (4193/915), (3308/885), (4905/1711), (1084/551), (138/133), (3847/707), (847407/152813), (35211085/6080747), (4645064/743515), (15759828/2239795), (129504763/15460639), (57309134/5354461), (51962535/3467611), (32134822/1352145), (20253919/443085), (4851479/37398), (25258555/530886), (5360669132/98787113), (119905448203/1987220773), (2901982675264/44933445971), (45205837486055/689931936553), (307078493073838/4963963519093), (104387653076331/1962402384031)]
[(2/1), (3/1), (4/1), (5/1), (6/1), (7/1), (8/1), (9/1), (10/1), (11/1), (12/1), (78/1), (77/1), (76/1), (75/1), (74/1), (73/1), (72/1), (71/1), (70/1), (69/1), (68/1), (67/1), (61/6), (4357/462), (50789/5852), (45089/5700), (39539/5550), (34137/5402), (3209/584), (2641/568), (18799/4970), (13969/4830), (9277/4692), (4721/4556), (24239/4087), (1600626/265777), (1380405411/221287673), (15189134204/2290025221), (684560325/93830209), (595266286/71039097), (1113163495/109545633), (1006391144/76274721), (759917189/40621221), (714638050/23873021), (7506827607/129590413), (7245441684/43796717), (6385155074/114432319), (408759419185/6466262269), (335553277583612/4782495219453)]
[(2/1), (3/1), (4/1), (5/1), (6/1), (7/1), (8/1), (9/1), (10/1), (11/1), (12/1), (13/1), (91/1), (90/1), (89/1), (88/1), (87/1), (86/1), (85/1), (84/1), (83/1), (82/1), (81/1), (80/1), (79/1), (145/13), (1217/117), (7718/801), (17337/1958), (5141/638), (9035/1247), (4690/731), (284/51), (1171/249), (12991/3403), (9670/3321), (643/324), (327/316), (14713/2291), (229541/35293), (31447476/4696403), (471197773/66903483), (6995384/918861), (12236231/1436565), (90316598/9080175), (2912729/237850), (13389561/831410), (350768873/15212461), (2333404792/62811485), (224540343/3108905)]
[(2/1), (3/1), (4/1), (5/1), (6/1), (7/1), (8/1), (9/1), (10/1), (11/1), (12/1), (13/1), (14/1), (105/1), (104/1), (103/1), (102/1), (101/1), (100/1), (99/1), (98/1), (97/1), (96/1), (95/1), (94/1), (93/1), (92/1), (85/7), (4141/364), (56759/5356), (51506/5253), (46355/5151), (8261/1010), (661/90), (2864/441), (26751/4753), (7365/1552), (1169/304), (2614/893), (8699/4371), (4421/4278), (27037/3910), (2459863/351985), (1683302712/235039019), (1560038824/208816361), (1359206763/170540045), (669643433/76587731), (54210676/5460521), (612020421/52060360), (2058279447/140460584)]
[(2/1), (3/1), (4/1), (5/1), (6/1), (7/1), (8/1), (9/1), (10/1), (11/1), (12/1), (13/1), (14/1), (15/1), (120/1), (119/1), (118/1), (117/1), (116/1), (115/1), (114/1), (113/1), (112/1), (111/1), (110/1), (109/1), (108/1), (107/1), (106/1), (197/15), (22052/1785), (81187/7021), (74284/6903), (33749/3393), (30414/3335), (18091/2185), (15944/2147), (5188/791), (4411/777), (2653/555), (2108/545), (8651/2943), (5762/2889), (5853/5671), (77354/10441), (8119065/1086061), (3416549738/447583931), (11928791159/1507723777), (5236551450/626752679), (1544605569/171073681)]
[(2/1), (3/1), (4/1), (5/1), (6/1), (7/1), (8/1), (9/1), (10/1), (11/1), (12/1), (13/1), (14/1), (15/1), (16/1), (136/1), (135/1), (134/1), (133/1), (132/1), (131/1), (130/1), (129/1), (128/1), (127/1), (126/1), (125/1), (124/1), (123/1), (122/1), (121/1), (113/8), (14401/1080), (226727/18090), (10995/938), (3357/308), (58019/5764), (12079/1310), (10789/1290), (123745/16512), (107489/16256), (91487/16002), (75737/15750), (60237/15500), (14995/5084), (9993/5002), (15217/14762), (108063/13673), (12961192/1627313), (1393557885/171847133)]
[(2/1), (3/1), (4/1), (5/1), (6/1), (7/1), (8/1), (9/1), (10/1), (11/1), (12/1), (13/1), (14/1), (15/1), (16/1), (17/1), (153/1), (152/1), (151/1), (150/1), (149/1), (148/1), (147/1), (146/1), (145/1), (144/1), (143/1), (142/1), (141/1), (140/1), (139/1), (138/1), (137/1), (257/17), (18497/1292), (154997/11476), (143672/11325), (132497/11175), (3283/298), (61/6), (2038/219), (89277/10585), (26279/3480), (2077/312), (5308/923), (48377/10011), (5501/1410), (4111/1390), (19186/9591), (9733/9453)]
[(2/1), (3/1), (4/1), (5/1), (6/1), (7/1), (8/1), (9/1), (10/1), (11/1), (12/1), (13/1), (14/1), (15/1), (16/1), (17/1), (18/1), (171/1), (170/1), (169/1), (168/1), (167/1), (166/1), (165/1), (164/1), (163/1), (162/1), (161/1), (160/1), (159/1), (158/1), (157/1), (156/1), (155/1), (154/1), (145/9), (2341/153), (41606/2873), (96917/7098), (89903/7014), (165945/13861), (10150/913), (4624/451), (62677/6683), (112151/13203), (99110/13041), (8623/1288), (7351/1272), (60949/12561), (48546/12403)]
[(2/1), (3/1), (4/1), (5/1), (6/1), (7/1), (8/1), (9/1), (10/1), (11/1), (12/1), (13/1), (14/1), (15/1), (16/1), (17/1), (18/1), (19/1), (190/1), (189/1), (188/1), (187/1), (186/1), (185/1), (184/1), (183/1), (182/1), (181/1), (180/1), (179/1), (178/1), (177/1), (176/1), (175/1), (174/1), (173/1), (172/1), (325/19), (58484/3591), (137327/8883), (128538/8789), (79895/5797), (14832/1147), (10273/851), (47156/4209), (171971/16653), (155500/16471), (13921/1629), (12310/1611), (107169/15931)]
[(2/1), (3/1), (4/1), (5/1), (6/1), (7/1), (8/1), (9/1), (10/1), (11/1), (12/1), (13/1), (14/1), (15/1), (16/1), (17/1), (18/1), (19/1), (20/1), (210/1), (209/1), (208/1), (207/1), (206/1), (205/1), (204/1), (203/1), (202/1), (201/1), (200/1), (199/1), (198/1), (197/1), (196/1), (195/1), (194/1), (193/1), (192/1), (191/1), (181/10), (36101/2090), (54973/3344), (51661/3312), (628951/42642), (586721/42230), (32053/2460), (4231/348), (66069/5858), (140627/13534), (127227/13400)]

151214(2)

Ruby


Euler brick(2)

http://mathworld.wolfram.com/EulerBrick.html

上記の記事を読むと、

Saunderson によって、Euler brickを作る方法が見つけられている。

（ただし、全てのEuler brickを作れるわけではない。）

ピタゴラス数から作るのだが、

ここでは、原始ピタゴラス数から作られるものだけを考え、

辺の組を出力してみる。

def f(a, b, c, n)
  return [] if 4 * a * b * c > n
  i, j, k = a * (4 * b * b - c * c).abs, b * (4 * a * a - c * c).abs, 4 * a * b * c
  l = Math.sqrt(i * i + j * j + k * k).to_i
  # 斜辺も整数かどうか？
  ary = [[i, j, k, l * l == i * i + j * j + k * k]]
  ary += f( a - 2 * b + 2 * c,  2 * a - b + 2 * c,  2 * a - 2 * b + 3 * c, n)
  ary += f( a + 2 * b + 2 * c,  2 * a + b + 2 * c,  2 * a + 2 * b + 3 * c, n)
  ary += f(-a + 2 * b + 2 * c, -2 * a + b + 2 * c, -2 * a + 2 * b + 3 * c, n)
  return ary
end

p f(3, 4, 5, 10 ** 7)

出力結果
[[117, 44, 240, false], [2035, 828, 3120, false], [11753, 10296, 16800, false], [42471, 54280, 59040, false], [117469, 194220, 161040, false], [272987, 550116, 371280, false], [561105, 1329328, 759360, false], [1052623, 2861136, 1419840, false], [1839941, 5637060, 2476080, false], [3039939, 10356940, 4084080, false], [4796857, 17980776, 6436320, false], [7285175, 29786328, 9765600, false], [1281735, 2229128, 5063520, false], [229229, 1274580, 2380560, false], [213785, 325008, 770880, false], [14715, 148148, 267120, false], [922077, 1721764, 2574000, false], [15939, 18460, 48720, false], [689481, 146960, 1110720, false], [5111703, 3736304, 7423680, false], [3455641, 3369960, 9653280, false], [157157, 593676, 942480, false], [3153645, 4106492, 5937360, false], [495, 4888, 8160, false], [274527, 7336, 480480, false], [2866149, 1651580, 4255440, false], [736255, 539352, 1815840, false], [27755, 42372, 62160, false], [3828995, 561132, 7046160, false], [201663, 186416, 262080, false], [851499, 580060, 799920, false], [2677103, 1458504, 1990560, false], [6956235, 3172148, 4302480, false], [15798015, 6209632, 8388480, false]]

151214

Ruby


Euler brick(1)

辺の長さが2500以下のprimitive Euler brickの辺の組を出力してみた。

class Array
  def gcd
    self.inject{|a, b| a.gcd(b)}
  end
end

n = 2500
(1..n).each{|i|
  (i..n).each{|j|
    (j..n).each{|k|
      l0 = Math.sqrt(i * i + j * j).to_i
      if l0 * l0 == i * i + j * j
        l1 = Math.sqrt(j * j + k * k).to_i
        if l1 * l1 == j * j + k * k
          l2 = Math.sqrt(i * i + k * k).to_i
          if l2 * l2 == i * i + k * k
            if [i, j, k].gcd == 1
              l3 = Math.sqrt(i * i + j * j + k * k).to_i
              # 斜辺も整数かどうか？
              p [i, j, k, l3 * l3 == i * i + j * j + k * k]
            end
          end
        end
      end
    }
  }
}

出力結果
[44, 117, 240, false]
[85, 132, 720, false]
[140, 480, 693, false]
[160, 231, 792, false]
[187, 1020, 1584, false]
[240, 252, 275, false]
[429, 880, 2340, false]
[1008, 1100, 1155, false]

151213(4)

Ruby


原始ピタゴラス数の和(2)

1≦x＜y＜z≦nなる原始ピタゴラス数について、
その和をf(n)とする。

1≦i≦7のとき、f(10 ** i) を求めてみた。

def f(a, b, c, n)
  return 0 if c > n
  s = a + b + c
  s += f( a - 2 * b + 2 * c,  2 * a - b + 2 * c,  2 * a - 2 * b + 3 * c, n)
  s += f( a + 2 * b + 2 * c,  2 * a + b + 2 * c,  2 * a + 2 * b + 3 * c, n)
  s += f(-a + 2 * b + 2 * c, -2 * a + b + 2 * c, -2 * a + 2 * b + 3 * c, n)
  return s
end

p (1..7).map{|i| f(3, 4, 5, 10 ** i)}

出力結果
[12, 1936, 179568, 18128932, 1809940702, 180861233118, 18090545150630]

151213(3)

Ruby


原始ピタゴラス数の和(1)

1≦x＜y＜z≦100なる原始ピタゴラス数について、

その和を求めてみる。

①Ruby学びはじめだった140315(2)分のコードに手を加えて求めてみた。

（本当に酷いコードです…。）

class Array
  def gcd
    self.inject{|a, b| a.gcd(b)}
  end
end

s = 0
for x in 1..98
  for y in x..99
    z = Math.sqrt(x**2+y**2)
    if z == z.to_i && z < 101 then
      z = z.to_i
      if [x, y, z].gcd == 1 then
        s += x + y + z
        puts "#{x} + #{y} + #{z} = #{x + y + z}"
      end
    end
  end
end
p s

出力結果
3 + 4 + 5 = 12
5 + 12 + 13 = 30
7 + 24 + 25 = 56
8 + 15 + 17 = 40
9 + 40 + 41 = 90
11 + 60 + 61 = 132
12 + 35 + 37 = 84
13 + 84 + 85 = 182
16 + 63 + 65 = 144
20 + 21 + 29 = 70
28 + 45 + 53 = 126
33 + 56 + 65 = 154
36 + 77 + 85 = 198
39 + 80 + 89 = 208
48 + 55 + 73 = 176
65 + 72 + 97 = 234
1936

②「ツイン・トライアングル」問題で紹介されていた
「Tree of primitive Pythagorean triples」を用いて求めてみた。

def f(a, b, c, n)
  return 0 if c > n
  s = a + b + c
  s += f( a - 2 * b + 2 * c,  2 * a - b + 2 * c,  2 * a - 2 * b + 3 * c, n)
  s += f( a + 2 * b + 2 * c,  2 * a + b + 2 * c,  2 * a + 2 * b + 3 * c, n)
  s += f(-a + 2 * b + 2 * c, -2 * a + b + 2 * c, -2 * a + 2 * b + 3 * c, n)
  return s
end

p f(3, 4, 5, 10 ** 2)

出力結果
1936

151213(2)

Ruby


Odd primes p such that Pi_{3,1}(p) = Pi_{3,2}(p) - 1

Pi_{m,n}(p) をmで割った余りがnとなるp以下の素数の個数とする。
Pi_{3,1}(p) = Pi_{3,2}(p) - 1 となる奇数の素数pについて、10^6まで調べてみた。
なお、オンライン整数列大辞典の
A098044（http://oeis.org/A098044/list）
にこの続きが載っている。

require 'prime'

def A098044(n)
  ary = []
  cnt0, cnt1, cnt2 = 0, 0, 0
  # たくさんの素数を用意
  Prime.each(10 ** 6).each{|i|
    if i > 3
      i % 3 == 1? cnt1 += 1 : cnt2 += 1
    elsif i == 3
      cnt0 += 1
    else
      cnt2 += 1
    end
    ary << i if cnt1 == cnt2 - 1 && i > 2
  }
  ary[0..n - 1]
end

p A098044(10)

出力結果
[3, 7, 13, 19, 37, 43, 79, 163, 223, 229]

151213

Ruby


Pythagorean prime

オンライン整数列大辞典の
A002144（http://oeis.org/A002144/list）
と比較し、答え合わせしてみる。

require 'prime'

def A002144(n)
  Prime.each(10 ** 6).select{|i| i % 4 == 1}[0..n - 1]
end
ary = A002144(54)

# OEIS A002144のデータ
ary0 =
[5,13,17,29,37,41,53,61,73,89,97,101,109,113,137,
 149,157,173,181,193,197,229,233,241,257,269,277,
 281,293,313,317,337,349,353,373,389,397,401,409,
 421,433,449,457,461,509,521,541,557,569,577,593,
 601,613,617]
# 一致の確認
p ary == ary0

151212(5)

Ruby


Hilbert prime(3)

Hilbert primeでないものを求めてみた。
オンライン整数列大辞典の
A054520（http://oeis.org/A054520/list）
と比較し、答え合わせしてみる。

def A054520(n)
  m = 10 ** 6
  list = Array.new(m + 1, false)
  1.step(m, 4){|i| list[i] = true}
  i = 5
  while i * i <= m
    if list[i]
      j = i + i
      while j <= m
        list[j] = false
        j += i
      end
    end
    i += 1
  end
  [1] + (5..m).select{|i| !list[i] && i % 4 == 1}[0..n - 1]
end
ary = A054520(52)

# OEIS A054520のデータ
ary0 =
[1,25,45,65,81,85,105,117,125,145,153,165,169,185,
 189,205,221,225,245,261,265,273,285,289,297,305,
 325,333,345,357,365,369,377,385,405,425,429,441,
 445,465,477,481,485,493,505,513,525,533,545,549,
 561,565,585]
# 一致の確認
p ary == ary0

151212(4)

Ruby


Hilbert prime(2)

もう少し調べてみた。

def A057948(n)
  # m以下のHilbert primeを探す
  m = 10 ** 6
  list = Array.new(m + 1, false)
  1.step(m, 4){|i| list[i] = true}
  i = 5
  while i * i <= m
    if list[i]
      j = i + i
      while j <= m
        list[j] = false
        j += i
      end
    end
    i += 1
  end
  (5..m).select{|i| list[i]}[0..n]
end

p A057948(10 ** 3)

出力結果
[5, 9, 13, 17, 21, 29, 33, 37, 41, 49, 53, 57, 61, 69, 73, 77, 89, 93, 97, 101, 109, 113, 121, 129, 133, 137, 141, 149, 157, 161, 173, 177, 181, 193, 197, 201, 209, 213, 217, 229, 233, 237, 241, 249, 253, 257, 269, 277, 281, 293, 301, 309, 313, 317, 321, 329, 337, 341, 349, 353, 361, 373, 381, 389, 393, 397, 401, 409, 413, 417, 421, 433, 437, 449, 453, 457, 461, 469, 473, 489, 497, 501, 509, 517, 521, 529, 537, 541, 553, 557, 569, 573, 577, 581, 589, 593, 597, 601, 613, 617, 633, 641, 649, 653, 661, 669, 673, 677, 681, 701, 709, 713, 717, 721, 733, 737, 749, 753, 757, 761, 769, 773, 781, 789, 797, 809, 813, 817, 821, 829, 849, 853, 857, 869, 877, 881, 889, 893, 913, 917, 921, 929, 933, 937, 941, 953, 961, 973, 977, 989, 993, 997, 1009, 1013, 1021, 1033, 1041, 1049, 1057, 1061, 1069, 1077, 1081, 1093, 1097, 1101, 1109, 1117, 1121, 1129, 1133, 1137, 1141, 1149, 1153, 1169, 1177, 1181, 1193, 1201, 1213, 1217, 1229, 1237, 1249, 1253, 1257, 1273, 1277, 1289, 1293, 1297, 1301, 1317, 1321, 1329, 1333, 1337, 1349, 1357, 1361, 1373, 1381, 1389, 1393, 1397, 1401, 1409, 1429, 1433, 1437, 1441, 1453, 1457, 1461, 1473, 1477, 1481, 1489, 1493, 1497, 1501, 1509, 1529, 1541, 1549, 1553, 1561, 1569, 1577, 1589, 1597, 1601, 1609, 1613, 1621, 1633, 1637, 1641, 1657, 1661, 1669, 1673, 1689, 1693, 1697, 1709, 1713, 1721, 1733, 1741, 1753, 1757, 1761, 1777, 1789, 1793, 1797, 1801, 1817, 1821, 1829, 1837, 1841, 1849, 1857, 1861, 1873, 1877, 1889, 1893, 1897, 1901, 1909, 1913, 1929, 1933, 1941, 1949, 1957, 1969, 1973, 1977, 1981, 1993, 1997, 2017, 2021, 2029, 2033, 2049, 2053, 2069, 2073, 2077, 2081, 2089, 2101, 2113, 2129, 2137, 2141, 2149, 2153, 2157, 2161, 2177, 2181, 2189, 2201, 2209, 2213, 2217, 2221, 2229, 2237, 2253, 2269, 2273, 2281, 2293, 2297, 2309, 2317, 2321, 2333, 2341, 2357, 2361, 2369, 2377, 2381, 2389, 2393, 2413, 2417, 2429, 2433, 2437, 2441, 2449, 2453, 2461, 2469, 2473, 2477, 2481, 2489, 2497, 2513, 2517, 2521, 2537, 2549, 2557, 2569, 2573, 2577, 2589, 2593, 2609, 2617, 2621, 2629, 2633, 2641, 2649, 2653, 2657, 2661, 2677, 2681, 2689, 2693, 2713, 2721, 2729, 2733, 2741, 2749, 2753, 2757, 2761, 2773, 2777, 2789, 2797, 2801, 2833, 2837, 2841, 2857, 2861, 2869, 2881, 2893, 2897, 2901, 2909, 2913, 2917, 2921, 2933, 2949, 2953, 2957, 2969, 2973, 2981, 3001, 3013, 3017, 3037, 3041, 3049, 3053, 3057, 3061, 3073, 3089, 3093, 3097, 3101, 3109, 3113, 3117, 3121, 3137, 3149, 3153, 3169, 3173, 3181, 3189, 3193, 3197, 3209, 3217, 3221, 3229, 3241, 3253, 3257, 3261, 3269, 3273, 3301, 3309, 3313, 3317, 3329, 3337, 3353, 3361, 3369, 3373, 3377, 3389, 3397, 3401, 3409, 3413, 3421, 3433, 3437, 3449, 3453, 3457, 3461, 3469, 3473, 3481, 3489, 3493, 3513, 3517, 3521, 3529, 3533, 3541, 3557, 3561, 3569, 3581, 3593, 3613, 3617, 3629, 3637, 3641, 3661, 3669, 3673, 3677, 3693, 3697, 3701, 3709, 3713, 3733, 3749, 3761, 3769, 3777, 3781, 3793, 3797, 3817, 3821, 3829, 3833, 3837, 3841, 3849, 3853, 3873, 3877, 3881, 3889, 3901, 3909, 3917, 3921, 3929, 3937, 3941, 3949, 3953, 3957, 3981, 3989, 3997, 4001, 4009, 4013, 4021, 4037, 4049, 4057, 4061, 4073, 4093, 4101, 4109, 4117, 4129, 4133, 4153, 4157, 4169, 4177, 4189, 4193, 4197, 4201, 4213, 4217, 4229, 4237, 4241, 4249, 4253, 4261, 4269, 4273, 4281, 4289, 4297, 4309, 4313, 4317, 4333, 4337, 4341, 4349, 4353, 4357, 4373, 4377, 4393, 4397, 4409, 4413, 4417, 4421, 4429, 4441, 4449, 4457, 4461, 4481, 4489, 4493, 4497, 4501, 4513, 4517, 4529, 4533, 4541, 4549, 4561, 4569, 4577, 4593, 4597, 4601, 4609, 4613, 4621, 4629, 4637, 4649, 4657, 4661, 4673, 4677, 4681, 4701, 4713, 4721, 4729, 4733, 4737, 4741, 4749, 4757, 4769, 4781, 4789, 4793, 4801, 4813, 4817, 4821, 4829, 4837, 4841, 4853, 4857, 4861, 4873, 4877, 4881, 4889, 4897, 4909, 4933, 4937, 4957, 4969, 4973, 4989, 4993, 4997, 5001, 5009, 5021, 5029, 5033, 5041, 5053, 5077, 5081, 5089, 5093, 5097, 5101, 5113, 5129, 5137, 5149, 5153, 5169, 5173, 5177, 5189, 5197, 5201, 5209, 5221, 5233, 5237, 5241, 5257, 5261, 5269, 5273, 5277, 5281, 5293, 5297, 5309, 5333, 5349, 5357, 5361, 5377, 5381, 5393, 5401, 5413, 5417, 5433, 5437, 5441, 5449, 5461, 5469, 5477, 5489, 5493, 5497, 5501, 5509, 5521, 5533, 5541, 5549, 5557, 5561, 5569, 5573, 5581, 5601, 5609, 5613, 5633, 5637, 5641, 5653, 5657, 5669, 5677, 5689, 5693, 5701, 5717, 5721, 5737, 5741, 5749, 5753, 5761, 5773, 5789, 5793, 5801, 5813, 5821, 5833, 5849, 5853, 5857, 5861, 5869, 5873, 5881, 5893, 5897, 5909, 5921, 5937, 5953, 5961, 5969, 5977, 5981, 5997, 6009, 6013, 6017, 6029, 6033, 6037, 6041, 6049, 6053, 6073, 6077, 6081, 6089, 6101, 6113, 6117, 6121, 6133, 6157, 6169, 6173, 6181, 6189, 6193, 6197, 6209, 6217, 6221, 6229, 6233, 6241, 6249, 6257, 6261, 6269, 6277, 6281, 6289, 6297, 6301, 6313, 6317, 6329, 6333, 6337, 6349, 6353, 6361, 6373, 6377, 6389, 6393, 6397, 6421, 6429, 6433, 6449, 6457, 6469, 6473, 6481, 6493, 6509, 6521, 6529, 6533, 6537, 6541, 6553, 6557, 6569, 6577, 6581, 6589, 6593, 6609, 6621, 6629, 6637, 6653, 6661, 6673, 6677, 6689, 6701, 6709, 6717, 6729, 6733, 6737, 6753, 6761, 6769, 6781, 6793, 6797, 6801, 6809, 6821, 6829, 6833, 6841, 6857, 6861, 6869, 6881, 6889, 6901, 6913, 6917, 6933, 6937, 6941, 6949, 6961, 6973, 6977, 6997, 7001, 7009, 7013, 7017, 7037, 7041, 7053, 7057, 7061, 7069, 7073, 7097, 7109, 7113, 7117, 7121, 7129, 7133, 7149, 7153, 7169, 7177, 7181, 7193, 7197, 7201, 7213, 7217, 7229, 7233, 7237, 7249, 7253, 7269, 7273, 7277, 7297, 7309, 7313, 7321, 7333, 7341, 7349, 7357, 7369, 7377, 7393, 7401, 7409, 7417, 7433, 7441, 7457, 7477, 7481, 7489, 7493, 7509, 7513, 7517, 7529, 7537, 7541, 7549, 7561, 7573, 7577, 7589, 7593, 7597, 7601, 7609, 7613, 7617, 7621, 7629, 7637, 7649, 7653, 7661, 7669, 7673, 7681, 7697, 7717, 7721, 7729, 7737, 7741, 7753, 7757, 7773, 7781, 7789, 7793, 7817, 7829, 7841, 7849, 7853, 7861, 7873, 7877, 7901, 7909, 7933, 7937, 7941, 7949, 7961, 7977, 7981, 7989, 7993]

151212(3)

Ruby


Hilbert prime(1)

オンライン整数列大辞典の
A057948（http://oeis.org/A057948/list）
と比較し、答え合わせしてみる。

=begin
p(=4k + 1)をHilbert primeとし、2p, 3p, …をfalseにしていく。
ちなみに、
2p, 3p, 4p, 6p, …は
2p = 4(2k) + 2,
3p = 4(3k) + 3,
4p = 4(4k + 1),
6p = 4(6k + 1) + 2,
…
なので、listを用意した時点でfalseになっている。
=end

def A057948(n)
  # m以下のHilbert primeを探す
  m = 10 ** 6
  list = Array.new(m + 1, false)
  1.step(m, 4){|i| list[i] = true}
  i = 5
  while i * i <= m
    if list[i]
      j = i + i
      while j <= m
        list[j] = false
        j += i
      end
    end
    i += 1
  end
  (5..m).select{|i| list[i]}[0..n]
end
ary = A057948(55)

# OEIS A057948のデータ
ary0 =
[5,9,13,17,21,29,33,37,41,49,53,57,61,69,73,77,89,
 93,97,101,109,113,121,129,133,137,141,149,157,161,
 173,177,181,193,197,201,209,213,217,229,233,237,
 241,249,253,257,269,277,281,293,301,309,313,317,
 321,329]
# 一致の確認
p ary == ary0