160409(2)

Ruby


Zagier's problems(3)

U(n) = 0 mod (6n + 2) のとき、u_n が整数となるので、

そのようなn を求めてみた。

require 'matrix'

def power(a, n, mod)
  return Matrix.I(a.row_size) if n == 0
  m = power(a, n >> 1, mod)
  m = (m * m).map{|i| i % mod}
  return m if n & 1 == 0
  (m * a).map{|i| i % mod}
end

def f(n)
  ary0 = [51, 10, 2]
  v = Vector.elements(ary0)
  ary1 =
  [[5, 2, 1],
   [1, 0, 0],
   [0, 1, 0]]
  a = Matrix[*ary1]
  mod = 6 * n + 2
  (power(a, n, mod) * v)[2] % mod
end

(1..10 ** 7).each{|i|
  p i if f(i) == 0
}

出力結果
2755452
4570452

160409

Ruby


Zagier's problems(2)

U(n) = 0 mod (6n + 2) のとき、u_n が整数となるので、

そのような最小のn を求めてみた。

（実行時間は40分くらい。）

require 'matrix'

def power(a, n, mod)
  return Matrix.I(a.row_size) if n == 0
  m = power(a, n >> 1, mod)
  m = (m * m).map{|i| i % mod}
  return m if n & 1 == 0
  (m * a).map{|i| i % mod}
end

def f(n)
  ary0 = [51, 10, 2]
  v = Vector.elements(ary0)
  ary1 =
  [[5, 2, 1],
   [1, 0, 0],
   [0, 1, 0]]
  a = Matrix[*ary1]
  mod = 6 * n + 2
  (power(a, n, mod) * v)[2] % mod
end

(1..10 ** 8).each{|i|
  if f(i) == 0
    p i
    break
  end
}

出力結果
2755452

160407

Ruby


Zagier's problems(1)

http://www-groups.dcs.st-and.ac.uk/~john/Zagier/Problems.html
に載っていた一見整数にならなさそうな数列を出力してみた。
ちなみに、U(n) はPerrin sequence と関係がある。

def f(n)
  return 1 if n <= 1
  (1..n).inject(:*)
end

def u(n)
  (0..n).inject(0){|s, k| s + f(2 * n + k).to_r / (f(2 * n - 2 * k) * f(3 * k + 1))}
end

def U(n)
  ((6 * n + 2) * u(n)).to_i
end

p (0..100).map{|i| u(i)}
p (0..100).map{|i| U(i)}

出力結果
[(1/1), (5/4), (51/14), (277/20), (1497/26), (4045/16), (43721/38), (118141/22), (638471/25), (6900995/56), (37295141/62), (201554637/68), (544632231/37), (1177345345/16), (15906858593/43), (85965651921/46), (929170680307/98), (5021529726405/104), (27137921296481/110), (73330918307761/58), (792606555396977/122), (2141747185756205/64), (11574673402485763/67), (125106329323279427/140), (676113834597852597/146), (3653931178440793365/152), (9873494945361475723/79), (106718925645094196557/164), (16963015858392872913/5), (1558448768553849869885/88), (16844691689554308253491/182), (91033996061172298426037/188), (491976261222077808376937/194), (53175879798445758939805/4), (14368956468116767650131161/206), (38827173290825245976506621/106), (209834219917204141506159391/109), (2268019848803459166617751555/224), (12257090470267762891066408581/230), (66241160488780141071579864797/236), (178994001616619845153324946351/121), (1934679427614026496567475601685/248), (5227807152512696002261128829313/127), (28252709191794126353026444694601/130), (305372999955606074035876437864947/266), (1650331450850232267596010225761765/272), (8918908672545961738757856893927921/278), (24100289632192939651508590679514041/142), (260491045117871552260197630808758017/290), (703887646395337740822890186827887085/148), (3804029566726753196026157155627707483/151), (41116337297966754475813329934397381187/308), (222205580049531460644817058668153510037/314), (240173726795408863713552853504163545505/64), (3244935333641125634302134392437396519443/163), (35073296184414876440801689518083754159517/332), (94773545445307964529892374713844587301441/169), (512185958744595824724565697479744087186165/172), (5536027065411604981806028163170902975226931/350), (29918426252927024136988188355201180399482197/356), (161688557312947522298002129497307196122237177/362), (436907833068001632372896526180054622192688605/184), (4722373871558838392461957709150361794570842601/374), (12760594623689573007049688389984667328873602301/190), (68962254823074705060083296185253753061131542711/193), (745387300597064181021493673121626561721380678915/392), (4028306711524998871361900927129117155509176770021/398), (21770232668465272128972658574259346407110908293357/404), (58826582032986711284304294199338296454147137842871/205), (635834592378322655972350160069030774511202371785445/416), (1718124761346012698563970317858383202389855659296033/211), (9285284981141372860076506043560285082914627806108481/214), (100361183449176102050993291867105414413216903070054387/434), (9861514667147964128952734984463363250954958870787235/8), (2931209470326325085284141716548756293004876751218815361/446), (7920587574233538791328251361450928418521323067475363521/226), (85610677989681176110943197471752281750025506915084563857/458), (231333474857833180402656567260556939358608851798271544365/232), (1250198639853080616915554285135987906961591088973917649203/235), (13512930975951820066877112317873859108800371803847347233347/476), (73028116388887089162853091864434361050565441078728949852277/482), (394666841176045247181850792528191499284351131179287279026485/488), (1066451684816976117150918629343850038815843454928870821035163/247), (11526878646910738755035740970359317747777702252726011718256877/500), (31147433407498821745482115080850090196718118107262414577225761/253), (168330497369221823599597235003953618770184136243966955425420845/256), (1819421585969124261732936551333295866236491537121445224281368371/518), (9832724786137506246554035926843993986656630466797618772262976757/524), (53139128097364223403435247206894468903296503680718918220728462217/530), (143590255822532626886008622219746814178016135437155636936224816485/268), (1552013539206192221913510752538101073573410992199791824575968066041/542), (4193783923709227920257511749388193546741208003214150294422734029181/274), (22664523413574864450087078121698815621457467143707698933625863028431/277), (48994476673958349680762867793415806295022582888426676467705106893415/112), (1323907578350676655659935030820966806954876856791926008015976592507461/566), (7154811705320116504007477988282389728967525047131312202624185757528317/572), (19333423033335863789880537171010480645111245931841273205745703753561791/289), (208967761322349547562480262717490552716002386269467282470721385643182005/584), (564663655225205654737965470129888537944712220060916408689606964493842753/295), (3051619460481713685042188150537943723084674732505890599124501911865957561/298), (32983816987040309016936223648616481935341598591572038908474168362278128627/602)]
[2, 10, 51, 277, 1497, 8090, 43721, 236282, 1276942, 6900995, 37295141, 201554637, 1089264462, 5886726725, 31813717186, 171931303842, 929170680307, 5021529726405, 27137921296481, 146661836615522, 792606555396977, 4283494371512410, 23149346804971526, 125106329323279427, 676113834597852597, 3653931178440793365, 19746989890722951446, 106718925645094196557, 576742539185357679042, 3116897537107699739770, 16844691689554308253491, 91033996061172298426037, 491976261222077808376937, 2658793989922287946990250, 14368956468116767650131161, 77654346581650491953013242, 419668439834408283012318782, 2268019848803459166617751555, 12257090470267762891066408581, 66241160488780141071579864797, 357988003233239690306649892702, 1934679427614026496567475601685, 10455614305025392004522257658626, 56505418383588252706052889389202, 305372999955606074035876437864947, 1650331450850232267596010225761765, 8918908672545961738757856893927921, 48200579264385879303017181359028082, 260491045117871552260197630808758017, 1407775292790675481645780373655774170, 7608059133453506392052314311255414966, 41116337297966754475813329934397381187, 222205580049531460644817058668153510037, 1200868633977044318567764267520817727525, 6489870667282251268604268784874793038886, 35073296184414876440801689518083754159517, 189547090890615929059784749427689174602882, 1024371917489191649449131394959488174372330, 5536027065411604981806028163170902975226931, 29918426252927024136988188355201180399482197, 161688557312947522298002129497307196122237177, 873815666136003264745793052360109244385377210, 4722373871558838392461957709150361794570842601, 25521189247379146014099376779969334657747204602, 137924509646149410120166592370507506122263085422, 745387300597064181021493673121626561721380678915, 4028306711524998871361900927129117155509176770021, 21770232668465272128972658574259346407110908293357, 117653164065973422568608588398676592908294275685742, 635834592378322655972350160069030774511202371785445, 3436249522692025397127940635716766404779711318592066, 18570569962282745720153012087120570165829255612216962, 100361183449176102050993291867105414413216903070054387, 542383306693138027092400424145484978802522737893297925, 2931209470326325085284141716548756293004876751218815361, 15841175148467077582656502722901856837042646134950727042, 85610677989681176110943197471752281750025506915084563857, 462666949715666360805313134521113878717217703596543088730, 2500397279706161233831108570271975813923182177947835298406, 13512930975951820066877112317873859108800371803847347233347, 73028116388887089162853091864434361050565441078728949852277, 394666841176045247181850792528191499284351131179287279026485, 2132903369633952234301837258687700077631686909857741642070326, 11526878646910738755035740970359317747777702252726011718256877, 62294866814997643490964230161700180393436236214524829154451522, 336660994738443647199194470007907237540368272487933910850841690, 1819421585969124261732936551333295866236491537121445224281368371, 9832724786137506246554035926843993986656630466797618772262976757, 53139128097364223403435247206894468903296503680718918220728462217, 287180511645065253772017244439493628356032270874311273872449632970, 1552013539206192221913510752538101073573410992199791824575968066041, 8387567847418455840515023498776387093482416006428300588845468058362, 45329046827149728900174156243397631242914934287415397867251726056862, 244972383369791748403814338967079031475112914442133382338525534467075, 1323907578350676655659935030820966806954876856791926008015976592507461, 7154811705320116504007477988282389728967525047131312202624185757528317, 38666846066671727579761074342020961290222491863682546411491407507123582, 208967761322349547562480262717490552716002386269467282470721385643182005, 1129327310450411309475930940259777075889424440121832817379213928987685506, 6103238920963427370084376301075887446169349465011781198249003823731915122, 32983816987040309016936223648616481935341598591572038908474168362278128627]

160405

Ruby


Göbel's Sequence(2)

オンライン整数列大辞典の
A005166 やA005167 のCOMMENTS を見ると、
「適当な素数p を見つけてきて、mod p で計算した結果a(p) は整数ではありません」

と示すのが楽そうなので、
そのようなp を見つけるコードを書いてみた。

もちろん、整数にならないような最小値（http://oeis.org/A108394/list）以上である。

require 'prime'
require 'OpenSSL'

def Gobel_prime(k)
  Prime.each(10 ** 7){|mod|
    x = 2
    (2..mod - 1).each{|i|
      x = OpenSSL::BN.new(i.to_s).mod_inverse(mod).to_i * x * (x ** (k - 1) + i - 1) % mod
    }
    return mod if x * (x ** (k - 1) + mod - 1) % mod > 0
  }
end

p (2..1000).map{|k| Gobel_prime(k)}

出力結果
[43, 89, 97, 251, 19, 239, 37, 79, 83, 239, 31, 431, 19, 79, 23, 827, 43, 173, 31, 103, 179, 73, 19, 431, 193, 101, 53, 811, 47, 1427, 19, 251, 29, 311, 137, 71, 23, 499, 43, 47, 19, 419, 31, 191, 83, 337, 59, 1559, 19, 127, 109, 163, 67, 353, 83, 191, 83, 107, 19, 503, 29, 191, 47, 83, 307, 1907, 19, 131, 37, 137, 31, 467, 31, 127, 47, 443, 19, 173, 31, 227, 23, 337, 83, 563, 19, 47, 167, 487, 29, 89, 83, 79, 137, 73, 19, 2039, 89, 311, 59, 127, 31, 173, 19, 239, 37, 71, 61, 167, 31, 457, 101, 179, 19, 173, 37, 179, 29, 191, 67, 563, 19, 103, 43, 151, 23, 101, 43, 239, 59, 139, 19, 47, 31, 541, 263, 101, 83, 647, 19, 179, 37, 103, 43, 839, 29, 83, 23, 167, 19, 167, 37, 331, 53, 167, 47, 167, 19, 211, 59, 1699, 31, 191, 31, 79, 43, 73, 19, 479, 23, 79, 47, 359, 29, 359, 19, 71, 37, 47, 97, 839, 61, 431, 53, 227, 19, 827, 37, 241, 383, 173, 23, 167, 19, 103, 97, 179, 47, 131, 31, 127, 29, 311, 19, 251, 53, 137, 43, 331, 79, 479, 19, 239, 23, 163, 47, 1427, 47, 347, 83, 307, 19, 251, 31, 47, 173, 101, 43, 83, 19, 229, 173, 751, 113, 191, 23, 101, 53, 73, 19, 1847, 61, 79, 47, 103, 59, 71, 19, 79, 37, 173, 31, 191, 31, 251, 83, 523, 19, 233, 31, 499, 47, 313, 47, 359, 19, 89, 109, 139, 43, 47, 67, 151, 59, 151, 19, 863, 109, 223, 23, 643, 31, 191, 19, 163, 29, 173, 53, 431, 31, 179, 43, 311, 19, 179, 37, 103, 101, 577, 113, 1559, 19, 127, 59, 331, 47, 227, 47, 179, 47, 73, 19, 227, 29, 167, 47, 47, 67, 179, 19, 79, 37, 167, 23, 491, 109, 79, 251, 131, 19, 479, 37, 163, 43, 193, 47, 101, 19, 223, 47, 379, 29, 137, 31, 311, 23, 103, 19, 563, 31, 439, 47, 127, 43, 89, 19, 337, 37, 167, 79, 479, 47, 47, 193, 251, 19, 239, 23, 211, 29, 389, 31, 383, 19, 179, 43, 107, 139, 563, 31, 467, 47, 73, 19, 659, 47, 71, 97, 727, 23, 233, 19, 83, 37, 197, 43, 503, 29, 79, 47, 419, 19, 821, 31, 79, 179, 139, 47, 47, 19, 227, 23, 439, 53, 677, 67, 419, 43, 269, 19, 179, 43, 151, 167, 151, 29, 101, 19, 103, 37, 317, 31, 503, 23, 439, 173, 101, 19, 683, 31, 167, 269, 89, 47, 251, 19, 163, 47, 103, 59, 239, 137, 127, 29, 47, 19, 71, 79, 131, 43, 419, 31, 251, 19, 367, 37, 461, 61, 179, 31, 349, 137, 179, 19, 83, 37, 79, 23, 127, 43, 167, 19, 79, 97, 557, 67, 167, 103, 167, 47, 233, 19, 263, 31, 163, 53, 179, 53, 137, 19, 47, 37, 311, 109, 251, 43, 197, 59, 431, 19, 659, 37, 311, 101, 317, 67, 1097, 19, 353, 47, 227, 23, 467, 31, 167, 179, 71, 19, 263, 31, 89, 107, 227, 71, 173, 19, 101, 29, 163, 47, 293, 61, 191, 23, 139, 19, 47, 37, 359, 53, 103, 31, 179, 19, 79, 83, 83, 97, 743, 31, 79, 317, 503, 19, 347, 23, 457, 89, 211, 47, 719, 19, 151, 37, 151, 83, 467, 113, 127, 97, 239, 19, 353, 31, 211, 43, 269, 23, 107, 19, 197, 47, 47, 29, 173, 67, 71, 101, 73, 19, 887, 53, 103, 71, 127, 43, 167, 19, 163, 23, 409, 31, 431, 31, 137, 47, 311, 19, 89, 31, 827, 29, 233, 59, 887, 19, 101, 43, 431, 47, 173, 23, 79, 107, 137, 19, 101, 47, 47, 263, 383, 31, 1049, 19, 487, 37, 167, 43, 167, 29, 179, 53, 103, 19, 167, 37, 163, 47, 353, 97, 383, 19, 191, 71, 179, 83, 71, 67, 241, 43, 73, 19, 1307, 31, 179, 23, 367, 29, 479, 19, 127, 37, 139, 47, 47, 59, 229, 257, 191, 19, 167, 37, 337, 149, 107, 223, 431, 19, 367, 109, 163, 31, 197, 31, 103, 29, 89, 19, 1931, 31, 79, 43, 251, 67, 479, 19, 79, 37, 131, 23, 83, 47, 151, 47, 151, 19, 101, 37, 719, 47, 47, 31, 191, 19, 103, 263, 71, 59, 563, 31, 677, 23, 73, 19, 647, 67, 193, 83, 211, 47, 137, 19, 307, 37, 103, 53, 947, 43, 227, 101, 479, 19, 479, 23, 227, 47, 173, 47, 599, 19, 137, 97, 167, 43, 701, 47, 47, 53, 211, 19, 1607, 61, 223, 83, 647, 23, 197, 19, 79, 29, 173, 31, 503, 31, 79, 43, 263, 19, 467, 31, 107, 59, 83, 71, 191, 19, 71, 23, 439, 53, 569, 47, 167, 47, 73, 19, 191, 29, 163, 179, 139, 31, 47, 19, 179, 37, 101, 61, 179, 23, 127, 89, 293, 19, 419, 37, 101, 43, 367, 223, 443, 19, 179, 227, 499, 29, 461, 79, 131, 107, 439, 19, 179, 31, 233, 59, 103, 43, 1163, 19, 307, 37, 163, 127, 251, 67, 79, 47, 47, 19, 347, 37, 79, 23, 151, 83, 71, 19, 173, 43, 787, 31, 89, 31, 83, 317, 73, 19, 359, 31, 197, 47, 499, 131, 257, 19, 373, 37, 197, 43, 863, 29, 631, 47, 379, 19, 107, 37, 103, 83, 193, 31, 311, 19, 47, 109, 463, 23, 281, 31, 643, 43, 191, 19, 443, 43, 101, 173, 467, 29, 617, 19, 127, 37, 347, 109, 101, 47, 103, 23, 199, 19, 443, 31, 79, 317, 71, 61, 1907, 19, 79, 47, 727, 47, 179, 107, 127, 29, 73, 19, 47, 23, 173, 43, 311, 53, 83, 19, 691, 37, 89, 31, 227, 31, 173, 59, 1439, 19, 131, 31, 163, 239, 127, 23, 167, 19, 167, 137, 211, 79, 227, 67, 229, 83, 191, 19, 2447, 47, 211, 47]

160404

整数列のLINKS の編集(1)

https://oeis.org/A181284
のLINKS を編集しました。
オズの数学には、20000 桁が世界記録だと載っていたので、

50000 桁求めてみた。

（実行時間は30分くらいだが、検証のため52000 桁を40分弱かけて求めた。）

160403(2)

Ruby


Alternating factorial(3)

(n - 1)! - (n - 2)! + (n - 3)! - … mod n が0 や2 の値になるn は少ない

のかもしれない。

今のところ、次のものしかわからない。

3612702! - 3612701! + 3612700! - 3612699! + … ≡ 0 mod 3612703

1108! - 1107! + 1106! - 1105! + … ≡ 2 mod 1109

def af(n, mod)
  a = 0
  f = 1
  (1..n).each{|i|
    f *= i
    f %= mod
    a = f - a
    a %= mod
  }
  a
end

p (1..100).map{|i| af(i - 1, i)}

h = {}
(1..100000).each{|i|
  j = af(i - 1, i)
  h.key?(j) ? h[j] = h[j].push(i) : h[j] = [i]
}
(0..3).each{|i| p [i, h[i]]}

出力結果
[0, 1, 1, 1, 4, 5, 3, 5, 7, 1, 4, 5, 12, 11, 4, 5, 8, 11, 14, 1, 10, 7, 18, 5, 19, 1, 16, 25, 19, 11, 9, 5, 4, 9, 24, 29, 36, 5, 25, 21, 1, 11, 5, 29, 34, 5, 6, 5, 10, 31, 25, 1, 20, 11, 4, 53, 52, 39, 32, 41, 48, 53, 52, 5, 64, 29, 5, 9, 64, 11, 64, 29, 15, 1, 19, 5, 59, 53, 4, 21, 16, 81, 61, 53, 59, 81, 19, 29, 47, 11, 38, 5, 40, 41, 14, 5, 57, 39, 70, 81]
[0, [1]]
[1, [2, 3, 4, 10, 20, 26, 41, 52, 74, 123, 130, 148, 260, 370, 740, 926, 962, 1852, 1924, 4630, 4810, 9260, 9620, 12038, 24076, 34262, 60190, 68524]]
[2, [1109]]
[3, [7, 262, 641, 1966, 2741, 3559, 4487, 19187, 22238, 24913, 45238]]

160403

Ruby


Alternating factorial(2)

「数論〈未解決問題〉の事典」の「B43」
に素因数分解が載っていたので試してみた。
（実行時間は5分くらい。）

require 'prime'

def af(n)
  a = 0
  f = 1
  (1..n).each{|i|
    f *= i
    a = f - a
    p [i, a, a.prime_division]
  }
end

af(20)

出力結果
[1, 1, []]
[2, 1, []]
[3, 5, [[5, 1]]]
[4, 19, [[19, 1]]]
[5, 101, [[101, 1]]]
[6, 619, [[619, 1]]]
[7, 4421, [[4421, 1]]]
[8, 35899, [[35899, 1]]]
[9, 326981, [[79, 1], [4139, 1]]]
[10, 3301819, [[3301819, 1]]]
[11, 36614981, [[13, 1], [2816537, 1]]]
[12, 442386619, [[29, 1], [15254711, 1]]]
[13, 5784634181, [[47, 1], [1427, 1], [86249, 1]]]
[14, 81393657019, [[23, 1], [73, 1], [211, 1], [229751, 1]]]
[15, 1226280710981, [[1226280710981, 1]]]
[16, 19696509177019, [[53, 1], [6581, 1], [56470483, 1]]]
[17, 335990918918981, [[47, 1], [7148742955723, 1]]]
[18, 6066382786809019, [[2683, 1], [2261044646593, 1]]]
[19, 115578717622022981, [[115578717622022981, 1]]]
[20, 2317323290554617019, [[8969, 1], [210101, 1], [1229743351, 1]]]

160402

Ruby


Alternating factorial(1)

昨日以下の問題を見かける。
http://integers.hatenablog.com/entry/2016/04/01/000130

その中の問２を解説してみる。

結論から言えば、実は、常にpn > n が成り立つわけでない。

https://en.wikipedia.org/wiki/Alternating_factorial
に載っているように、
3612702! - 3612701! + 3612700! - 3612699! + … ≡ 0 mod 3612703
より、n ≧ 3612702 のとき、
n! - (n - 1)! + (n - 2)! - … ≡ 0 mod 3612703
よって、3612703 ≧ pn となる。
したがって、n ≧ 3612703 のとき、
n ≧ pn となる。

念のため、
3612702! - 3612701! + 3612700! - 3612699! + … ≡ 0 mod 3612703
を確認しておく。

def af(n, mod)
  a = 0
  f = 1
  (1..n).each{|i|
    f *= i
    f %= mod
    a = f - a
    a %= mod
    p [i, a] if i <= 10 || i > 3612690
  }
end

mod = 3612703
af(3612750, mod)

出力結果
[1, 1]
[2, 1]
[3, 5]
[4, 19]
[5, 101]
[6, 619]
[7, 4421]
[8, 35899]
[9, 326981]
[10, 3301819]
[3612691, 2384440]
[3612692, 1884506]
[3612693, 2391173]
[3612694, 2480152]
[3612695, 1901684]
[3612696, 3552494]
[3612697, 3462171]
[3612698, 1204237]
[3612699, 1806349]
[3612700, 2]
[3612701, 3612702]
[3612702, 0]
[3612703, 0]
[3612704, 0]
[3612705, 0]
[3612706, 0]
[3612707, 0]
[3612708, 0]
[3612709, 0]
[3612710, 0]
[3612711, 0]
[3612712, 0]
[3612713, 0]
[3612714, 0]
[3612715, 0]
[3612716, 0]
[3612717, 0]
[3612718, 0]
[3612719, 0]
[3612720, 0]
[3612721, 0]
[3612722, 0]
[3612723, 0]
[3612724, 0]
[3612725, 0]
[3612726, 0]
[3612727, 0]
[3612728, 0]
[3612729, 0]
[3612730, 0]
[3612731, 0]
[3612732, 0]
[3612733, 0]
[3612734, 0]
[3612735, 0]
[3612736, 0]
[3612737, 0]
[3612738, 0]
[3612739, 0]
[3612740, 0]
[3612741, 0]
[3612742, 0]
[3612743, 0]
[3612744, 0]
[3612745, 0]
[3612746, 0]
[3612747, 0]
[3612748, 0]
[3612749, 0]
[3612750, 0]

160401(2)

新たな整数列(1)

https://oeis.org/A270599

が追加されました。

160401

OEIS のOFFSET

OFFSET の欄には通常二つの数が記載されている。
一つ目の数は「初項が第何項であるか」を表している。
二つ目の数は「絶対値が 1 より大きな数が初めて現れるのは初項から数えて何項目か」を表している。