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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A068585 Numbers k such that the denominator of (Sum_{j=1..k} 1/j)^3 equals the denominator of Sum_{j=1..k} 1/j^3.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 10, 11, 12, 13, 14, 15, 16, 17, 27, 28, 29, 30, 31, 32, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 289, 290, 291, 292, 293, 1004, 1005, 1006, 1007, 1008, 1009, 1010, 1011, 1012, 1013
Offset: 1

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Author

Benoit Cloitre, Mar 27 2002

Keywords

Formula

Numbers k such that A002805(k)^3 = A007409(k).

A303988 Triangle read by rows: numerators of c_{n,k}, n >= 0, 0 <= k <= n, used in the proof that Zeta(3) is irrational.

Original entry on oeis.org

0, 1, 5, 9, 29, 115, 251, 65, 5191, 1039, 2035, 10391, 2077, 72703, 58157, 256103, 259703, 1817471, 1817521, 7270009, 1454021, 28567, 67323, 25243, 389467, 21810107, 47982293, 6854599, 9822481, 9895981, 11132213, 66793523, 11755653433, 2351131157, 30564700141, 30564710941, 78708473, 237497419, 237487619, 23511313481, 23511309071, 61129406407, 5557218637, 61129406447, 244517610353
Offset: 0

Views

Author

Wolfdieter Lang, May 16 2018

Keywords

Comments

The corresponding denominators are given in A303989.
The numerators of the rational triangle c_{n,k} are denoted by T(n,k). The triangle c_{n,k} is used to compute Apéry's sequence of rationals a_n = A059415(n)/A059416(n), satisfying a certain three term recurrence, as a(n) = Sum_{k=0..n} c_{n,k}*(binomial(n+k,k)*binomial(n,k))^2 = Sum_{k=0..n} (T(n,k)/A303989(n,k))*A303987(n,k).
The column k = 0 gives the numerators of Zeta3(n) = A007408(n)/A007409(n), with Zeta3(0) := 0.

Examples

			The triangle T(n, k) begins:
  n/k      0       1        2        3           4          5           6
  0:       0
  1:       1       5
  2:       9      29      115
  3:     251      65     5191     1039
  4:    2035   10391     2077    72703       58157
  5:  256103  259703  1817471  1817521     7270009    1454021
  6:   28567   67323    25243   389467    21810107   47982293     6854599
  ...
  row n = 7: 9822481 9895981 11132213 66793523 11755653433 2351131157 30564700141 30564710941,
  row n = 8: 78708473 237497419 237487619 23511313481 23511309071 61129406407 5557218637 61129406447 244517610353,
  row n = 9: 19148110939 19237016539 211601625329 211601801729 2750823224027 42320357851 550164649543 550164651163 37411196140169 37411196579209,
  ...
------------------------------------------------------------------------------
The rational triangle c_{n,k} starts:
  n\k        0            1              2                3               4
  0:        0/1
  1:        1/1          5/4
  2:        9/8         29/24         115/96
  3:      251/216       65/54        5191/4320        1039/864
  4:    2035/1728    10391/8640      2077/1728      72703/60480      58157/48384
  ...
  row n = 5:  256103/216000 259703/216000 1817471/1512000 1817521/1512000 7270009/6048000 1454021/1209600,
  ...

References

  • Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 137-153.

Links

Crossrefs

Cf. A005259, A007408/A007409, A059415, A059416, A063007, A303987, A303989.

Programs

  • PARI
    T(n,k) = numerator(sum(m=1, n, 1/m^3) + sum(m=1, k, (-1)^(m-1)/(2*m^3*binomial(n,m)*binomial(n+m,m)))) \\ Jason Yuen, May 27 2025

Formula

T(n,k) = numerator(c_{n,k}), with c_{n,k} = Zeta3(n) + Sum_{m=1..k} (-1)^(m-1)/(2*m^3*B(n,m)), where Zeta3(n) = Sum_{m=1..n} 1/m^3 = A007408(n)/A007409(n) and B(n,m) = A063007(n,m).

A303989 Triangle read by rows: denominators of c_{n,k}, n >= 0, k = 0..n, used in the proof that Zeta(3) is irrational.

Original entry on oeis.org

1, 1, 4, 8, 24, 96, 216, 54, 4320, 864, 1728, 8640, 1728, 60480, 48384, 216000, 216000, 1512000, 1512000, 6048000, 1209600, 24000, 56000, 21000, 324000, 18144000, 39916800, 5702400, 8232000, 8232000, 9261000, 55566000, 9779616000, 1955923200, 25427001600, 25427001600, 65856000, 197568000, 197568000, 19559232000, 19559232000, 50854003200, 4623091200, 50854003200, 203416012800
Offset: 0

Views

Author

Wolfdieter Lang, May 16 2018

Keywords

Comments

See A303988 for details, references and links.

Examples

			The triangle T(n, k) begins:
  n\k       0       1       2        3         4          5          6
  0:        1
  1:        1       4
  2:        8      24      96
  3:      216      54    4320      864
  4:     1728    8640    1728    60480      48384
  5:   216000  216000 1512000  1512000    6048000    1209600
  6:    24000   56000   21000   324000   18144000   39916800     5702400
  ...
  row n = 7: 8232000 8232000 9261000 55566000 9779616000 1955923200 25427001600 25427001600,
  row n = 8: 65856000 197568000 197568000 19559232000 19559232000 50854003200 4623091200 50854003200 203416012800,
  row n = 9: 16003008000 16003008000 176033088000 176033088000 2288430144000 35206617600 457686028800 457686028800 31122649958400 31122649958400,
  ...
For the first rationals c_{n,k} see A303988.

Crossrefs

Cf. A007408/A007409, A063007, A303988.

Programs

  • PARI
    T(n,k) = denominator(sum(m=1, n, 1/m^3) + sum(m=1, k, (-1)^(m-1)/(2*m^3*binomial(n,m)*binomial(n+m,m)))) \\ Jason Yuen, May 28 2025

Formula

T(n, k) = denominator(c_{n,k}), with c_{n,k} = Zeta3(n) + Sum_{m=1..k} (-1)^(m-1)/(2*m^3*B(n, m)), where Zeta3(n) = Sum_{m=1..n} 1/m^3 = A007408(n)/A007409(n) and B(n, m) = A063007(n, m).

A322266 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = denominator of Sum_{j=1..n} 1/j^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 6, 1, 1, 8, 36, 12, 1, 1, 16, 216, 144, 60, 1, 1, 32, 1296, 1728, 3600, 20, 1, 1, 64, 7776, 20736, 216000, 3600, 140, 1, 1, 128, 46656, 248832, 12960000, 24000, 176400, 280, 1, 1, 256, 279936, 2985984, 777600000, 12960000, 8232000, 705600, 2520, 1
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 01 2018

Keywords

Examples

			Square array begins:
  1,       1,          1,              1,                  1,  ...
  2,     3/2,        5/4,            9/8,              17/16,  ...
  3,    11/6,      49/36,        251/216,          1393/1296,  ...
  4,   25/12,    205/144,      2035/1728,        22369/20736,  ...
  5,  137/60,  5269/3600,  256103/216000,  14001361/12960000,  ...

Links

Crossrefs

Columns k=0..10 give A000012, A002805, A007407, A007409, A007480, A069052, A103346, A103348, A103350, A103352, A103717.
Numerators are in A322265.
Cf. A103438, A276487.

Programs

  • Mathematica
    Table[Function[k, Denominator[Sum[1/j^k, {j, 1, n}]]][i - n], {i, 0, 10}, {n, 1, i}] // Flatten
Table[Function[k, Denominator[HarmonicNumber[n, k]]][i - n], {i, 0, 10}, {n, 1, i}] // Flatten
Table[Function[k, Denominator[SeriesCoefficient[PolyLog[k, x]/(1 - x), {x, 0, n}]]][i - n], {i, 0, 10}, {n, 1, i}] // Flatten

Formula

G.f. of column k: PolyLog(k,x)/(1 - x), where PolyLog() is the polylogarithm function (for rationals Sum_{j=1..n} 1/j^k).
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