A276485 -id:A276485 - cp's OEIS frontend

A276487 Denominator of Sum_{k=1..n} 1/k^n.

1, 4, 216, 20736, 777600000, 46656000000, 768464444160000000, 247875891108249600000000, 4098310578334288576512000000000, 413109706296096288512409600000000, 7425496288284402957501110551810198732800000000000

Offset: 1

Ilya Gutkovskiy, Sep 05 2016

Also denominator of zeta(n) - Hurwitz zeta(n,n+1), where zeta(s) is the Riemann zeta function and Hurwitz zeta(s,a) is the Hurwitz zeta function.

Sum_{k>=1} 1/k^n = zeta(n).

			1, 5/4, 251/216, 22369/20736, 806108207/777600000, 47464376609/46656000000, 774879868932307123/768464444160000000, ...
a(3) = 216, because 1/1^3 + 1/2^3 + 1/3^3 = 251/216.

Eric Weisstein's World of Mathematics, Harmonic Number
Eric Weisstein's World of Mathematics, Riemann Zeta Function
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function

Cf. A001008, A002805, A007406, A007407, A031971, A276485 (numerators).

A276487:=n->denom(add(1/k^n, k=1..n)): seq(A276487(n), n=1..12); # Wesley Ivan Hurt, Sep 07 2016

Table[Denominator[HarmonicNumber[n, n]], {n, 1, 11}]

a(n) = denominator(sum(k=1, n, 1/k^n)); \\ Michel Marcus, Sep 06 2016

A322265 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = numerator of Sum_{j=1..n} 1/j^k.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 5, 11, 4, 1, 9, 49, 25, 5, 1, 17, 251, 205, 137, 6, 1, 33, 1393, 2035, 5269, 49, 7, 1, 65, 8051, 22369, 256103, 5369, 363, 8, 1, 129, 47449, 257875, 14001361, 28567, 266681, 761, 9, 1, 257, 282251, 3037465, 806108207, 14011361, 9822481, 1077749, 7129, 10

Offset: 1

Ilya Gutkovskiy, Dec 01 2018

			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,  ...

Eric Weisstein's World of Mathematics, Harmonic Number

Columns k=0..10 give A000027, A001008, A007406, A007408, A007410, A099828, A103345, A103347, A103349, A103351, A103716.
Denominators are in A322266.
Cf. A103438, A276485.

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

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).

cp's OEIS Frontend

A276487 Denominator of Sum_{k=1..n} 1/k^n.

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