201110

PARI


A008298 とA075525

Product_{k>0} (1 - x^k)^(-u)

= 1 + u*x/1! + (3*u+u^2)*x^2/2! + (8*u+9*u^2+u^3)*x^3/3! + ...

のx^k/k!の係数部分すなわち

 1, u, 3*u+u^2, 8*u+9*u^2+u^3, ...

を出力してみた。

同様に

Product_{k>0} (1 + x^k)^u

= 1 + u*x/1! + (u+u^2)*x^2/2! + (8*u+3*u^2+u^3)*x^3/3! + ...

のx^k/k!の係数部分すなわち

 1, u, u+u^2, 8*u+3*u^2+u^3, ...

も出力してみた。

N=10; x='x+O('x^N);
a(n) = sigma(n);
Vec(serlaplace(exp(sum(n=1, N, u*a(n)*x^n/n))))

a(n) = sumdiv(n, d, (-1)^(d+1)*n/d);
Vec(serlaplace(exp(sum(n=1, N, u*a(n)*x^n/n))))

出力結果

%3 = [1, u, u^2 + 3*u, u^3 + 9*u^2 + 8*u, u^4 + 18*u^3 + 59*u^2 + 42*u, u^5 + 30*u^4 + 215*u^3 + 450*u^2 + 144*u, u^6 + 45*u^5 + 565*u^4 + 2475*u^3 + 3394*u^2 + 1440*u, u^7 + 63*u^6 + 1225*u^5 + 9345*u^4 + 28294*u^3 + 30912*u^2 + 5760*u, u^8 + 84*u^7 + 2338*u^6 + 27720*u^5 + 147889*u^4 + 340116*u^3 + 293292*u^2 + 75600*u, u^9 + 108*u^8 + 4074*u^7 + 69552*u^6 + 579369*u^5 + 2341332*u^4 + 4335596*u^3 + 3032208*u^2 + 524160*u]

%5 = [1, u, u^2 + u, u^3 + 3*u^2 + 8*u, u^4 + 6*u^3 + 35*u^2 + 6*u, u^5 + 10*u^4 + 95*u^3 + 110*u^2 + 144*u, u^6 + 15*u^5 + 205*u^4 + 585*u^3 + 1594*u^2 + 480*u, u^7 + 21*u^6 + 385*u^5 + 1995*u^4 + 8974*u^3 + 8064*u^2 + 5760*u, u^8 + 28*u^7 + 658*u^6 + 5320*u^5 + 35329*u^4 + 70252*u^3 + 125292*u^2 + 5040*u, u^9 + 36*u^8 + 1050*u^7 + 12096*u^6 + 110649*u^5 + 392364*u^4 + 1178540*u^3 + 684144*u^2 + 524160*u]

201104

Ruby


ゴロム定規

最短ゴロム定規を出力してみた。

def is_Golomb_ruler(ary)
  n = ary.size
  a = []
  (0..n - 2).each{|i|
    (i + 1..n - 1).each{|j|
      k = ary[j] - ary[i]
      if a.include?(k)
        return false
      else
        a << k
      end
    }
  }
  true
end

# 対称型を含む
def A(deg, l)
  a = []
  (1..l).to_a.combination(deg - 1){|c|
    ary = [0] + c
    a << ary if is_Golomb_ruler(ary)
  }
  a
end

# 対称型を省く
def B(deg, l)
  a = []
  A(deg, l).each{|ary|
    if !a.include?(ary.map{|i| ary[-1] - i}.reverse)
      a << ary
      p ary
    end
  }
end

B(2,  1)
B(3,  3)
B(4,  6)
B(5, 11)
B(6, 17)
B(7, 25)

出力結果

[0, 1]

[0, 1, 3]

[0, 1, 4, 6]

[0, 1, 4, 9, 11]

[0, 2, 7, 8, 11]

[0, 1, 4, 10, 12, 17]

[0, 1, 4, 10, 15, 17]

[0, 1, 8, 11, 13, 17]

[0, 1, 8, 12, 14, 17]

[0, 1, 4, 10, 18, 23, 25]

[0, 1, 7, 11, 20, 23, 25]

[0, 1, 11, 16, 19, 23, 25]

[0, 2, 3, 10, 16, 21, 25]

[0, 2, 7, 13, 21, 22, 25]

201020(2)

Crystal


A006950(2)

先程と同様にCrystal に書き換えてみた。

def a006950(n)
  return 1 if n < 3
  a = (1..n - 1).map{|i| [i]}
  ary = [[n]]
  while a.size != 0
    b = Array(Array(Int32)).new
    a.each{|i|
      x = i[-1].to_i
      s = n - i.sum
      t = x % 2
      (x + t..s - 1).each{|j| b << i.clone + [j]}
      ary << i.clone + [s] if s >= x + t
    }
    a = b
  end
  ary.size
end

p (0..90).map{|i| a006950(i)}

出力結果

[1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 21, 28, 35, 43, 55, 70, 86, 105, 130, 161, 196, 236, 287, 350, 420, 501, 602, 722, 858, 1016, 1206, 1431, 1687, 1981, 2331, 2741, 3206, 3740, 4368, 5096, 5922, 6868, 7967, 9233, 10670, 12306, 14193, 16357, 18803, 21581, 24766, 28394, 32490, 37128, 42411, 48406, 55159, 62777, 71416, 81182, 92152, 104494, 118434, 134131, 151725, 171467, 193676, 218607, 246491, 277691, 312677, 351841, 395537, 444314, 498852, 559727, 627509, 703002, 787171, 880896, 985043, 1100775, 1229494, 1372499, 1531086, 1706958, 1902121, 2118469, 2357947, 2623017]

201020

Crystal


A316384(2)

Crystal ではinject ではなく、reduce だが、

和の場合はsum が使える。

def f(ary)
  f_ary = Array(Int32).new
  while !(ary.all?{|i| i == 0})
    a = Array(Int32).new
    s = 0
    ary.each{|i|
      if i == 1
        a << 0
        s += 1
      elsif i > 0
        a << i - 2
        s += 2
      end
    }
    ary = a
    f_ary << s
  end
  f_ary
end

def a316384(n)
  return 1 if n < 3
  a = (1..n - 1).map{|i| [i]}
  ary = Array(Array(Int32)).new
  while a.size != 0
    b = Array(Array(Int32)).new
    a.each{|i|
      x = i[-1].to_i
      s = n - i.sum
      t = x % 2
      (x + t..s - 1).each{|j| b << i.clone + [j]}
      ary << i.clone + [s] if s >= x + t
    }
    a = b
  end
  ary.select{|i| i.reverse == f(i)}.size
end

p (0..90).map{|i| a316384(i)}

出力結果

[1, 1, 1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 4, 2, 5, 2, 5, 2, 6, 3, 8, 4, 9, 4, 10, 4, 12, 6, 15, 7, 17, 7, 19, 8, 22, 10, 26, 12, 30, 13, 33, 14, 38, 17, 45, 21, 51, 22, 56, 24, 64, 29, 74, 33, 83, 36, 92, 40, 104, 46, 119, 53, 133, 58, 147, 63, 165, 73, 187, 83, 208, 90, 229, 99, 256, 113, 288, 127, 319, 138, 351, 152, 390, 171, 435, 191, 481]

201016

Crystal


Number of solutions to +- 1 +- 2 +- 3 +- ... +- n = 0(6)

以下を用いて、n が4000 のときの値を求めることはおすすめしない。

require "big"

def a063865(n)
  ary = [1.to_big_i]
  m = n * (n + 1) // 4
  a = Array.new(m + 1){0.to_big_i}
  a[0] = 1.to_big_i
  (1..n).each{|i|
    b = a.clone
    (0..[(i - 1) * i / 2, m - i].min).each{|k| b[k + i] += a[k]}
    a = b
    i * (i + 1) % 4 == 0 ? ary << a[i * (i + 1) // 4] : ary << 0.to_big_i
  }
  ary
end

p a063865(1000)[-1]

出力結果

467708586991053780130476928496471502490480200263913521598374850751872554493810444457551220080062611698129340084492921358002542871554381676735263500063986630523649271451391806356678763700553308953903873511563867155215734010997332808966175771715884247295296277348179194597363883854664431808932677416

201010

Python


足し算と引き算だけで円を描く(3)

Python で書いてみた。

def A(r):
    xy = [['□' for _ in range(r + 1)] for _ in range(r + 1)]
    a, x, y = r, 0, r
    while x <= y:
        xy[x][y], xy[y][x] = '■', '■'
        a, x = a - x - x - 1, x + 1
        if a < 0:
            a, y = a + y + y - 1, y - 1 
    for i in xy:
        print(''.join(i))

for i in range(21):
    A(i)

出力結果

■

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201009

Ruby


足し算と引き算だけで円を描く(2)

文字コードの設定が変わると出力がかわる。

chcp で932 の場合は

chcp 65001 で変更すればよい。

200920

Ruby

足し算と引き算だけで円を描く(1)

原点を中心とする半径r の円にみえる座標を選ぶことを考える。

(r^2 - r ≦) x^2 + y^2 ≦ r^2 + r

を満たす(x, y) を探す。

つまり

a(x, y) = r^2 + r - x^2 - y^2

が0 以上(、2r 以下)となるように(x, y) を探す。

(0, r) から出発したとき、a(x, y) = r である。

yを固定し、xを1増加させ、右に移動することを考える。

このとき、a(x, y) は2x+1 減少する。

右に移動し続け、a(x, y) < 0 となったとき、

y を1減少させ、下へ移動させる。

このとき、a(x, y) は2y-1 増加する。

この作業を繰り返し、

円の8分の1を描き、

あとは対称性を用いて全体を表示させる。

このアルゴリズムを使い、円の一部を描いてみた。

def A(r)
  xy = Array.new(r + 1){Array.new(r + 1, '□')}
  a, x, y = r, 0, r
  while x <= y
    xy[x][y], xy[y][x] = '■', '■'
    a, x = a - x - x - 1, x + 1
    a, y = a + y + y - 1, y - 1 if a < 0
  end
  xy.each{|i| p i.join}
end

(0..20).each{|i| A(i)}

出力結果

"■"

"□■"

"■■"

"□□■"

"□□■"

"■■□"

"□□□■"

"□□□■"

"□□■□"

"■■□□"

"□□□□■"

"□□□□■"

"□□□□■"

"□□□■□"

"■■■□□"

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"■■■□□□"

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200906

PARI


A010034

本当に最大公約数が1 にならないことを確かめてみた。

f(n) = gcd(n^17+9, (n+1)^17+9);
f(8424432925592889329288197322308900672459420460792433)

出力結果

%2 = 8936582237915716659950962253358945635793453256935559

200830

Ruby


88888

意外なことに88888 が見つからなかった。

(1..9).to_a.permutation{|i|
  a, b = i[0..4].join.to_i, i[5..-1].join.to_i
  if (a - b) % 11111 == 0
    puts "#{a} - #{b} = #{a - b}"
  elsif (a + b) % 11111 == 0
    puts "#{a} + #{b} = #{a + b}"
  end
}

出力結果

13579 - 2468 = 11111

13597 - 2486 = 11111

13759 - 2648 = 11111

13795 - 2684 = 11111

13957 - 2846 = 11111

13975 - 2864 = 11111

15379 - 4268 = 11111

15397 - 4286 = 11111

15739 - 4628 = 11111

15793 - 4682 = 11111

15937 - 4826 = 11111

15973 - 4862 = 11111

17359 - 6248 = 11111

17395 - 6284 = 11111

17539 - 6428 = 11111

17593 - 6482 = 11111

17935 - 6824 = 11111

17953 - 6842 = 11111

19357 - 8246 = 11111

19375 - 8264 = 11111

19537 - 8426 = 11111

19573 - 8462 = 11111

19735 - 8624 = 11111

19753 - 8642 = 11111

25718 - 3496 = 22222

25871 - 3649 = 22222

27158 - 4936 = 22222

27185 - 4963 = 22222

28571 - 6349 = 22222

28715 - 6493 = 22222

31478 - 9256 = 22222

31487 - 9265 = 22222

31748 - 9526 = 22222

31784 - 9562 = 22222

31847 - 9625 = 22222

31874 - 9652 = 22222

41268 - 7935 = 33333

41286 - 7953 = 33333

46279 - 1835 = 44444

46297 - 1853 = 44444

46819 - 2375 = 44444

46981 - 2537 = 44444

47629 - 3185 = 44444

47962 - 3518 = 44444

48169 - 3725 = 44444

48196 - 3752 = 44444

49627 - 5183 = 44444

49681 - 5237 = 44444

49762 - 5318 = 44444

49816 - 5372 = 44444

51238 - 6794 = 44444

51427 - 6983 = 44444

52138 - 7694 = 44444

53416 - 8972 = 44444

54127 - 9683 = 44444

54316 - 9872 = 44444

56789 - 1234 = 55555

56798 - 1243 = 55555

56834 - 1279 = 55555

56879 - 1324 = 55555

56897 - 1342 = 55555

56942 - 1387 = 55555

56978 - 1423 = 55555

56987 - 1432 = 55555

57239 - 1684 = 55555

57248 - 1693 = 55555

57689 - 2134 = 55555

57698 - 2143 = 55555

57869 - 2314 = 55555

57896 - 2341 = 55555

57941 - 2386 = 55555

57968 - 2413 = 55555

57986 - 2431 = 55555

58319 - 2764 = 55555

58346 - 2791 = 55555

58679 - 3124 = 55555

58697 - 3142 = 55555

58724 - 3169 = 55555

58769 - 3214 = 55555

58796 - 3241 = 55555

58967 - 3412 = 55555

58976 - 3421 = 55555

59417 - 3862 = 55555

59426 - 3871 = 55555

59678 - 4123 = 55555

59687 - 4132 = 55555

59723 - 4168 = 55555

59768 - 4213 = 55555

59786 - 4231 = 55555

59831 - 4276 = 55555

59867 - 4312 = 55555

59876 - 4321 = 55555

61289 - 5734 = 55555

61298 - 5743 = 55555

61379 - 5824 = 55555

61397 - 5842 = 55555

61478 - 5923 = 55555

61487 - 5932 = 55555

69153 - 2487 = 66666

69513 - 2847 = 66666

71358 - 4692 = 66666

71529 - 4863 = 66666

71934 - 5268 = 66666

73158 - 6492 = 66666

73194 - 6528 = 66666

73491 - 6825 = 66666

74931 - 8265 = 66666

75129 - 8463 = 66666

81256 - 3479 = 77777

81472 - 3695 = 77777

81526 - 3749 = 77777

81742 - 3965 = 77777

82156 - 4379 = 77777

82516 - 4739 = 77777

84172 - 6395 = 77777

84712 - 6935 = 77777

85126 - 7349 = 77777

85216 - 7439 = 77777

87142 - 9365 = 77777

87412 - 9635 = 77777

91234 + 8765 = 99999

91235 + 8764 = 99999

91243 + 8756 = 99999

91246 + 8753 = 99999

91253 + 8746 = 99999

91256 + 8743 = 99999

91264 + 8735 = 99999

91265 + 8734 = 99999

91324 + 8675 = 99999

91325 + 8674 = 99999

91342 + 8657 = 99999

91347 + 8652 = 99999

91352 + 8647 = 99999

91357 + 8642 = 99999

91374 + 8625 = 99999

91375 + 8624 = 99999

91423 + 8576 = 99999

91426 + 8573 = 99999

91432 + 8567 = 99999

91437 + 8562 = 99999

91462 + 8537 = 99999

91467 + 8532 = 99999

91473 + 8526 = 99999

91476 + 8523 = 99999

91523 + 8476 = 99999

91526 + 8473 = 99999

91532 + 8467 = 99999

91537 + 8462 = 99999

91562 + 8437 = 99999

91567 + 8432 = 99999

91573 + 8426 = 99999

91576 + 8423 = 99999

91624 + 8375 = 99999

91625 + 8374 = 99999

91642 + 8357 = 99999

91647 + 8352 = 99999

91652 + 8347 = 99999

91657 + 8342 = 99999

91674 + 8325 = 99999

91675 + 8324 = 99999

91734 + 8265 = 99999

91735 + 8264 = 99999

91743 + 8256 = 99999

91746 + 8253 = 99999

91753 + 8246 = 99999

91756 + 8243 = 99999

91764 + 8235 = 99999

91765 + 8234 = 99999

92134 + 7865 = 99999

92135 + 7864 = 99999

92143 + 7856 = 99999

92146 + 7853 = 99999

92153 + 7846 = 99999

92156 + 7843 = 99999

92164 + 7835 = 99999

92165 + 7834 = 99999

92314 + 7685 = 99999

92315 + 7684 = 99999

92341 + 7658 = 99999

92348 + 7651 = 99999

92351 + 7648 = 99999

92358 + 7641 = 99999

92384 + 7615 = 99999

92385 + 7614 = 99999

92413 + 7586 = 99999

92416 + 7583 = 99999

92431 + 7568 = 99999

92438 + 7561 = 99999

92461 + 7538 = 99999

92468 + 7531 = 99999

92483 + 7516 = 99999

92486 + 7513 = 99999

92513 + 7486 = 99999

92516 + 7483 = 99999

92531 + 7468 = 99999

92538 + 7461 = 99999

92561 + 7438 = 99999

92568 + 7431 = 99999

92583 + 7416 = 99999

92586 + 7413 = 99999

92614 + 7385 = 99999

92615 + 7384 = 99999

92641 + 7358 = 99999

92648 + 7351 = 99999

92651 + 7348 = 99999

92658 + 7341 = 99999

92684 + 7315 = 99999

92685 + 7314 = 99999

92834 + 7165 = 99999

92835 + 7164 = 99999

92843 + 7156 = 99999

92846 + 7153 = 99999

92853 + 7146 = 99999

92856 + 7143 = 99999

92864 + 7135 = 99999

92865 + 7134 = 99999

93124 + 6875 = 99999

93125 + 6874 = 99999

93142 + 6857 = 99999

93147 + 6852 = 99999

93152 + 6847 = 99999

93157 + 6842 = 99999

93174 + 6825 = 99999

93175 + 6824 = 99999

93214 + 6785 = 99999

93215 + 6784 = 99999

93241 + 6758 = 99999

93248 + 6751 = 99999

93251 + 6748 = 99999

93258 + 6741 = 99999

93284 + 6715 = 99999

93285 + 6714 = 99999

93412 + 6587 = 99999

93417 + 6582 = 99999

93421 + 6578 = 99999

93428 + 6571 = 99999

93471 + 6528 = 99999

93478 + 6521 = 99999

93482 + 6517 = 99999

93487 + 6512 = 99999

93512 + 6487 = 99999

93517 + 6482 = 99999

93521 + 6478 = 99999

93528 + 6471 = 99999

93571 + 6428 = 99999

93578 + 6421 = 99999

93582 + 6417 = 99999

93587 + 6412 = 99999

93714 + 6285 = 99999

93715 + 6284 = 99999

93741 + 6258 = 99999

93748 + 6251 = 99999

93751 + 6248 = 99999

93758 + 6241 = 99999

93784 + 6215 = 99999

93785 + 6214 = 99999

93824 + 6175 = 99999

93825 + 6174 = 99999

93842 + 6157 = 99999

93847 + 6152 = 99999

93852 + 6147 = 99999

93857 + 6142 = 99999

93874 + 6125 = 99999

93875 + 6124 = 99999

94123 + 5876 = 99999

94126 + 5873 = 99999

94132 + 5867 = 99999

94137 + 5862 = 99999

94162 + 5837 = 99999

94167 + 5832 = 99999

94173 + 5826 = 99999

94176 + 5823 = 99999

94213 + 5786 = 99999

94216 + 5783 = 99999

94231 + 5768 = 99999

94238 + 5761 = 99999

94261 + 5738 = 99999

94268 + 5731 = 99999

94283 + 5716 = 99999

94286 + 5713 = 99999

94312 + 5687 = 99999

94317 + 5682 = 99999

94321 + 5678 = 99999

94328 + 5671 = 99999

94371 + 5628 = 99999

94378 + 5621 = 99999

94382 + 5617 = 99999

94387 + 5612 = 99999

94612 + 5387 = 99999

94617 + 5382 = 99999

94621 + 5378 = 99999

94628 + 5371 = 99999

94671 + 5328 = 99999

94678 + 5321 = 99999

94682 + 5317 = 99999

94687 + 5312 = 99999

94713 + 5286 = 99999

94716 + 5283 = 99999

94731 + 5268 = 99999

94738 + 5261 = 99999

94761 + 5238 = 99999

94768 + 5231 = 99999

94783 + 5216 = 99999

94786 + 5213 = 99999

94823 + 5176 = 99999

94826 + 5173 = 99999

94832 + 5167 = 99999

94837 + 5162 = 99999

94862 + 5137 = 99999

94867 + 5132 = 99999

94873 + 5126 = 99999

94876 + 5123 = 99999

95123 + 4876 = 99999

95126 + 4873 = 99999

95132 + 4867 = 99999

95137 + 4862 = 99999

95162 + 4837 = 99999

95167 + 4832 = 99999

95173 + 4826 = 99999

95176 + 4823 = 99999

95213 + 4786 = 99999

95216 + 4783 = 99999

95231 + 4768 = 99999

95238 + 4761 = 99999

95261 + 4738 = 99999

95268 + 4731 = 99999

95283 + 4716 = 99999

95286 + 4713 = 99999

95312 + 4687 = 99999

95317 + 4682 = 99999

95321 + 4678 = 99999

95328 + 4671 = 99999

95371 + 4628 = 99999

95378 + 4621 = 99999

95382 + 4617 = 99999

95387 + 4612 = 99999

95612 + 4387 = 99999

95617 + 4382 = 99999

95621 + 4378 = 99999

95628 + 4371 = 99999

95671 + 4328 = 99999

95678 + 4321 = 99999

95682 + 4317 = 99999

95687 + 4312 = 99999

95713 + 4286 = 99999

95716 + 4283 = 99999

95731 + 4268 = 99999

95738 + 4261 = 99999

95761 + 4238 = 99999

95768 + 4231 = 99999

95783 + 4216 = 99999

95786 + 4213 = 99999

95823 + 4176 = 99999

95826 + 4173 = 99999

95832 + 4167 = 99999

95837 + 4162 = 99999

95862 + 4137 = 99999

95867 + 4132 = 99999

95873 + 4126 = 99999

95876 + 4123 = 99999

96124 + 3875 = 99999

96125 + 3874 = 99999

96142 + 3857 = 99999

96147 + 3852 = 99999

96152 + 3847 = 99999

96157 + 3842 = 99999

96174 + 3825 = 99999

96175 + 3824 = 99999

96214 + 3785 = 99999

96215 + 3784 = 99999

96241 + 3758 = 99999

96248 + 3751 = 99999

96251 + 3748 = 99999

96258 + 3741 = 99999

96284 + 3715 = 99999

96285 + 3714 = 99999

96412 + 3587 = 99999

96417 + 3582 = 99999

96421 + 3578 = 99999

96428 + 3571 = 99999

96471 + 3528 = 99999

96478 + 3521 = 99999

96482 + 3517 = 99999

96487 + 3512 = 99999

96512 + 3487 = 99999

96517 + 3482 = 99999

96521 + 3478 = 99999

96528 + 3471 = 99999

96571 + 3428 = 99999

96578 + 3421 = 99999

96582 + 3417 = 99999

96587 + 3412 = 99999

96714 + 3285 = 99999

96715 + 3284 = 99999

96741 + 3258 = 99999

96748 + 3251 = 99999

96751 + 3248 = 99999

96758 + 3241 = 99999

96784 + 3215 = 99999

96785 + 3214 = 99999

96824 + 3175 = 99999

96825 + 3174 = 99999

96842 + 3157 = 99999

96847 + 3152 = 99999

96852 + 3147 = 99999

96857 + 3142 = 99999

96874 + 3125 = 99999

96875 + 3124 = 99999

97134 + 2865 = 99999

97135 + 2864 = 99999

97143 + 2856 = 99999

97146 + 2853 = 99999

97153 + 2846 = 99999

97156 + 2843 = 99999

97164 + 2835 = 99999

97165 + 2834 = 99999

97314 + 2685 = 99999

97315 + 2684 = 99999

97341 + 2658 = 99999

97348 + 2651 = 99999

97351 + 2648 = 99999

97358 + 2641 = 99999

97384 + 2615 = 99999

97385 + 2614 = 99999

97413 + 2586 = 99999

97416 + 2583 = 99999

97431 + 2568 = 99999

97438 + 2561 = 99999

97461 + 2538 = 99999

97468 + 2531 = 99999

97483 + 2516 = 99999

97486 + 2513 = 99999

97513 + 2486 = 99999

97516 + 2483 = 99999

97531 + 2468 = 99999

97538 + 2461 = 99999

97561 + 2438 = 99999

97568 + 2431 = 99999

97583 + 2416 = 99999

97586 + 2413 = 99999

97614 + 2385 = 99999

97615 + 2384 = 99999

97641 + 2358 = 99999

97648 + 2351 = 99999

97651 + 2348 = 99999

97658 + 2341 = 99999

97684 + 2315 = 99999

97685 + 2314 = 99999

97834 + 2165 = 99999

97835 + 2164 = 99999

97843 + 2156 = 99999

97846 + 2153 = 99999

97853 + 2146 = 99999

97856 + 2143 = 99999

97864 + 2135 = 99999

97865 + 2134 = 99999

98234 + 1765 = 99999

98235 + 1764 = 99999

98243 + 1756 = 99999

98246 + 1753 = 99999

98253 + 1746 = 99999

98256 + 1743 = 99999

98264 + 1735 = 99999

98265 + 1734 = 99999

98324 + 1675 = 99999

98325 + 1674 = 99999

98342 + 1657 = 99999

98347 + 1652 = 99999

98352 + 1647 = 99999

98357 + 1642 = 99999

98374 + 1625 = 99999

98375 + 1624 = 99999

98423 + 1576 = 99999

98426 + 1573 = 99999

98432 + 1567 = 99999

98437 + 1562 = 99999

98462 + 1537 = 99999

98467 + 1532 = 99999

98473 + 1526 = 99999

98476 + 1523 = 99999

98523 + 1476 = 99999

98526 + 1473 = 99999

98532 + 1467 = 99999

98537 + 1462 = 99999

98562 + 1437 = 99999

98567 + 1432 = 99999

98573 + 1426 = 99999

98576 + 1423 = 99999

98624 + 1375 = 99999

98625 + 1374 = 99999

98642 + 1357 = 99999

98647 + 1352 = 99999

98652 + 1347 = 99999

98657 + 1342 = 99999

98674 + 1325 = 99999

98675 + 1324 = 99999

98734 + 1265 = 99999

98735 + 1264 = 99999

98743 + 1256 = 99999

98746 + 1253 = 99999

98753 + 1246 = 99999

98756 + 1243 = 99999

98764 + 1235 = 99999

98765 + 1234 = 99999