A099828 -id:A099828 - cp's OEIS frontend

A103716 Numerators of sum_{k=1..n} 1/k^10 =: Zeta(10,n).

1, 1025, 60526249, 61978938025, 605263128567754849, 605263138567754849, 170971856382109814342232401, 175075181098169912564190119249, 10338014371627802833957102351534201, 413520574906423083987893722912609

Offset: 1

Wolfdieter Lang, Feb 15 2005

a(n) gives the partial sums, Zeta(10,n), of Euler's Zeta(10). Zeta(k,n) is also called H(k,n) because for k=1 these are the harmonic numbers H(n) = A001008/A002805.

For the denominators see A103717 and for the rationals Zeta(10,n) see the W. Lang link under A103345.

For k=1..9 see: A001008/A002805, A007406/A007407, A007408/A007409, A007410/A007480, A099828/A069052, A103345/A103346, A103347/A103348, A103349/A103350, A103351/A103352.

s=0;lst={};Do[s+=n^1/n^11;AppendTo[lst,Numerator[s]],{n,3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 24 2009 *)
Table[ HarmonicNumber[n, 10] // Numerator, {n, 1, 10}] (* Jean-François Alcover, Dec 04 2013 *)

a(n) = numerator(sum_{k=1..n} 1/k^10).

G.f. for rationals Zeta(10, n): polylogarithm(10, x)/(1-x).

A125503 Smallest number k such that the numerator of the generalized harmonic number H(k,n) = Sum_{i=1..k} 1/i^n is a prime.

Original entry on oeis.org

2, 2, 3, 2, 23, 73, 15, 2, 3, 5, 13, 57, 3, 171, 5, 2, 21, 7, 55, 8902, 26, 1298, 115, 139, 3, 2019, 3, 4, 3, 15, 56, 177

Offset: 1

Alexander Adamchuk, Dec 28 2006, Jan 31 2007

a(n) = 2 for n = {1,2,4,8,16,...}. Corresponding Fermat primes A019434.
a(n) = 3 for n = {3,9,13,25,27,29,95,107,153,159,...}.
a(n) = 5 for n = {10,15,60,90,197,209,...}.
a(n) = 7 for n = {18,47,112,155,273,...}.
a(n) = 15 for n = {7,30,43,...}.
a(21) = 26. a(28) = 4. a(31) = 56. a(144) = 9.
From Alexander Adamchuk, Apr 18 2010: (Start)
a(22)-a(25) = {1298,115,139,3}.
a(27)-a(32) = {3,4,3,15,56,177}.
a(n) = 3 for all n>2 listed in A125706. (End)
a(26) = 2019. - Alexander Adamchuk, Apr 26 2010
a(20) > 3000. - Michael S. Branicky, Jun 25 2022

Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A001008, A007406, A007408, A007410, A099828, A019434, A125706.

Do[n = 1; f = 0; While[Not[PrimeQ[Numerator[f]]], f = f + 1/n^x; n++ ]; Print[{x, n - 1}], {x, 1, 25}] (* Alexander Adamchuk, Apr 18 2010 *)

a(n) = my(k=1); while (!ispseudoprime(numerator(sum(i=1, k, 1/i^n))), k++); k; \\ Michel Marcus, Jun 04 2022

from sympy import isprime
from fractions import Fraction
def a(n):
    Hkn, k = Fraction(1, 1), 1
    while not isprime(Hkn.numerator):
        k += 1
        Hkn += Fraction(1, k**n)
    return k
print([a(n) for n in range(1, 20)]) # Michael S. Branicky, Jun 11 2022

a(22)-a(25) from Alexander Adamchuk, Apr 18 2010
a(26)-a(32) from Alexander Adamchuk, Apr 26 2010
Incorrect a(20) removed by Michael S. Branicky, Jun 25 2022
a(20) = 8902 from Michael S. Branicky, Jun 12 2023

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

A103716 Numerators of sum_{k=1..n} 1/k^10 =: Zeta(10,n).

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A125503 Smallest number k such that the numerator of the generalized harmonic number H(k,n) = Sum_{i=1..k} 1/i^n is a prime.

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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.

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