A007410 -id:A007410 - cp's OEIS frontend

A103349 Numerators of sum_{k=1..n} 1/k^8 = Zeta(8,n).

1, 257, 1686433, 431733409, 168646292872321, 168646392872321, 972213062238348973121, 248886558707571775009601, 1632944749460578249437992161, 1632944765723715465050248417

Offset: 1

Wolfdieter Lang, Feb 15 2005

a(n) gives the partial sums, Zeta(8,n) of Euler's Zeta(8). 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 A103350 and for the rationals Zeta(8,n) see the W. Lang link under A103345.

For k=1..7 see: A001008/A002805, A007406/A007407, A007408/A007409, A007410/A007480, A099828/A069052, A103345/A103346, A103347/A103348.

s=0;lst={};Do[s+=n^1/n^9;AppendTo[lst,Numerator[s]],{n,3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 24 2009 *)
Table[ HarmonicNumber[n, 8] // Numerator, {n, 1, 10}] (* Jean-François Alcover, Dec 04 2013 *)
Accumulate[1/Range[10]^8]//Numerator (* Harvey P. Dale, Aug 11 2024 *)

a(n)=numerator(sum_{k=1..n} 1/k^8).

G.f. for rationals Zeta(8, n): polylogarithm(8, x)/(1-x).

A128670 Least number k > 0 such that k^n does not divide the denominator of generalized harmonic number H(k,n) nor the denominator of alternating generalized harmonic number H'(k,n).

Original entry on oeis.org

77, 20, 94556602, 42, 444, 20, 104, 42, 76, 20, 77, 110, 3504, 20, 903, 42, 1107, 20, 104, 42, 77, 20, 2948, 110, 136, 20, 76, 42, 903, 20, 77, 42, 268, 20, 7004, 110, 1752, 20, 19203, 42, 77, 20, 104, 42, 76, 20, 370, 110, 1107, 20, 77, 42, 12246, 20, 104, 42

Offset: 1

Alexander Adamchuk, Mar 24 2007

nonn

Generalized harmonic numbers are defined as H(m,k) = Sum_{j=1..m}1/j^k. Alternating generalized harmonic numbers are defined as H'(m,k) = Sum_{j=1..m} (-1)^(j+1)/j^k.
Some apparent periodicity in {a(n)} (not without exceptions): a(n) = 20 for n = 2 + 4m, a(n) = 42 for n = 4 + 12m and 8 + 12m, a(n) = 76 for n = 9 + 18m, a(n) = 77 for n = 1 + 10m, a(n) = 104 for n = 7 + 12m, a(n) = 110 for n = 12m, a(n) = 136 for n = 25 + 32m, etc.
See more details in Comments at A128672 and A125581.

Max Alekseyev, Table of n, a(n) for n=1..158.
Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A001008, A002805, A058313, A058312, A007406, A007407, A119682, A007410, A120296, A125581, A126196, A126197, A128672, A128673, A128674, A128675, A128676, A128671, A128670.

More terms and b-file from Max Alekseyev, May 07 2010

A128674 Numbers m such that m^k does not divide the denominator of the m-th generalized harmonic number H(m,k) nor the denominator of the m-th alternating generalized harmonic number H'(m,k), for k = 4.

Original entry on oeis.org

42, 110, 156, 272, 294, 342, 506, 812, 930, 1210, 1332, 1640, 1806, 2028, 2058, 2162, 2756, 3422, 3660, 4422, 4624, 4970, 5256, 6162, 6498, 6806, 7832, 9312, 10100, 10506, 11342, 11638, 11772, 12656, 13310, 14406, 16002, 17030, 18632, 19182, 22052, 22650, 23548, 24492, 26364

Offset: 1

Alexander Adamchuk, Mar 20 2007

nonn

Generalized harmonic numbers are defined as H(m,k) = Sum_{j=1..m} 1/j^k. Alternating generalized harmonic numbers are defined as H'(m,k) = Sum_{j=1..m} (-1)^(j+1)/j^k.

Sequence contains geometric progressions of the form (p-1)*p^k for k > 0 and some prime p > 5. Note the factorization of initial terms of {a(n)} = {6*7, 10*11, 12*13, 16*17, 6*7^2, 18*19, 22*23, 28*29, 30*31, 10*11*2, 36*37, 40*41, 42*43, 12*13^2, 6*7^3, ...}. See more details in Comments at A128672 and A125581.

Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A001008, A002805, A058313, A058312, A007406, A007407, A119682, A007410, A120296, A125581, A126196, A126197, A128672, A128675, A128676.

k=4; f=0; g=0; Do[ f=f+1/n^k; g=g+(-1)^(n+1)*1/n^k; kf=Denominator[f]; kg=Denominator[g]; If[ !IntegerQ[kf/n^k] && !IntegerQ[kg/n^k], Print[n] ], {n,1,2000} ]

Edited and extended by Max Alekseyev, May 09 2010

A103351 Numerators of sum_{k=1..n} 1/k^9 = Zeta(9,n).

Original entry on oeis.org

1, 513, 10097891, 5170139875, 10097934603139727, 373997614931101, 15092153145114981831307, 7727182467755471289426059, 4106541588424891370931874221019, 4106541592523201949266162797531

Offset: 1

Wolfdieter Lang, Feb 15 2005

a(n) gives the partial sums, Zeta(9,n), of Euler's Zeta(9). 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 A103352 and for the rationals Zeta(9,n) see the W. Lang link under A103345.

For k=1..8 see: A001008/A002805, A007406/A007407, A007408/A007409, A007410/A007480, A099828/A069052, A103345/A103346, A103347/A103348, A103349/A103350.

s=0;lst={};Do[s+=n^1/n^10;AppendTo[lst,Numerator[s]],{n,3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 24 2009 *)
Table[ HarmonicNumber[n, 9] // Numerator, {n, 1, 10}] (* Jean-François Alcover, Dec 04 2013 *)

a(n) = numerator(sum_{k=1..n} 1/k^9).

G.f. for rationals Zeta(9, n): polylogarithm(9, x)/(1-x).

A120290 Numerator of generalized harmonic number H(p-1,2p)= Sum[ 1/k^(2p), {k,1,p-1}] divided by p^2 for prime p>3.

Original entry on oeis.org

2479157521, 159936660724017234488561, 1119583852472161859174156302552583713828739479026834819554843860744244189

Offset: 3

Alexander Adamchuk, Jul 08 2006

Generalized harmonic number is H(n,m)= Sum[ 1/k^m, {k,1,n} ]. The numerator of generalized harmonic number H(p-1,2p) is divisible by p^2 for prime p>3.

			With prime(3) = 5, a(3) = numerator[ 1 + 1/2^10 + 1/3^10 + 1/4^10 ] / 5^2 = 61978938025 / 25 = 2479157521.

Alexander Adamchuk, First 5 terms.
Eric Weisstein's World of Mathematics, Harmonic Number.
Eric Weisstein's World of Mathematics, Wolstenholme's Theorem.

Cf. A119722, A099828, A099827, A001008, A007406, A007408, A007410.

Table[Numerator[Sum[1/k^(2*Prime[n]),{k,1,Prime[n]-1}]],{n,3,7}]/Table[Prime[n]^2,{n,3,7}]

a(n) = numerator[ Sum[ 1/k^(2*Prime[n]), {k,1,Prime[n]-1} ]] / Prime[n]^2 for n>2.

A128671 Least number k > 0 such that k^p does not divide the denominator of generalized harmonic number H(k,p) nor the denominator of alternating generalized harmonic number H'(k,p), where p = prime(n).

Original entry on oeis.org

20, 94556602, 444, 104, 77, 3504, 1107, 104, 2948, 903, 77, 1752, 77, 104, 370

Offset: 1

Alexander Adamchuk, Mar 24 2007, Mar 26 2007

Generalized harmonic numbers are defined as H(m,k) = Sum_{i=1..m} 1/i^k. Alternating generalized harmonic numbers are defined as H'(m,k) = Sum_{i=1..m} (-1)^(i+1)*1/i^k.
a(18)..a(24) = {77,104,77,136,104,370,136}. a(26)..a(27) = {77,104}.
a(n) is currently unknown for n = {16,17,25,...}. See more details in Comments at A128672 and A125581.

			a(2) = A128673(1) = 94556602.

Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A128670, A001008, A002805, A058313, A058312, A007406, A007407, A119682, A007410, A120296, A125581, A126196, A126197, A128672, A128673, A128674, A128675, A128676.

a(n) = A128670(prime(n)).

a(9) = 2948 and a(12) = 1752 from Max Alekseyev

Edited by Max Alekseyev, Feb 20 2019

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

Original entry on oeis.org

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).

A130416 Numerator of partial sums for a series of (17/18)Zeta(4) = (17/1680)Pi^4.

Original entry on oeis.org

1, 49, 6623, 741857, 13247611, 3060203141, 13645449045719, 218327192834879, 100212182125865461, 1904031462407822767, 2534265876944902342877, 58288115171766608401171, 128058989033214718801833487

Offset: 1

Wolfdieter Lang, Jul 13 2007

Denominators are given by A130417.

The rationals r(n) = 2*Sum_{k=1..n} 1/(k^4*binomial(2*k,k)) tend, in the limit n->infinity, to (18/17)*Zeta(4) = (17/1680)*Pi^4, approximately 1.022194166.

			Rationals: 1, 49/48, 6623/6480, 741857/725760, 13247611/12960000, ...

L. Berggren, T. Borwein and P. Borwein, Pi: A Source Book, Springer, New York, 1997, p. 687.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 89, Exercise.

W. Lang, Rationals and limit.
A. van der Poorten, A proof that Euler missed ... Apery's proof of the irrationality of zeta(3). An informal report, Math. Intelligencer 1 (1978/79), no. 4, 195-203; reprinted in Pi: A Source Book, pp. 439-447, footnote 10, p. 446 (conjecture).

Partial sums for Zeta(4): A007410/A007480.

a(n) = numerator(r(n)), n >= 1, with the rationals defined above.

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

A103349 Numerators of sum_{k=1..n} 1/k^8 = Zeta(8,n).

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A128670 Least number k > 0 such that k^n does not divide the denominator of generalized harmonic number H(k,n) nor the denominator of alternating generalized harmonic number H'(k,n).

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A128674 Numbers m such that m^k does not divide the denominator of the m-th generalized harmonic number H(m,k) nor the denominator of the m-th alternating generalized harmonic number H'(m,k), for k = 4.

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A103351 Numerators of sum_{k=1..n} 1/k^9 = Zeta(9,n).

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A120290 Numerator of generalized harmonic number H(p-1,2p)= Sum[ 1/k^(2p), {k,1,p-1}] divided by p^2 for prime p>3.

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A128671 Least number k > 0 such that k^p does not divide the denominator of generalized harmonic number H(k,p) nor the denominator of alternating generalized harmonic number H'(k,p), where p = prime(n).

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

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A130416 Numerator of partial sums for a series of (17/18)*Zeta(4) = (17/1680)*Pi^4.

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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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A130416 Numerator of partial sums for a series of (17/18)Zeta(4) = (17/1680)Pi^4.