A103346 -id:A103346 - cp's OEIS frontend

A103345 Numerator of Sum_{k=1..n} 1/k^6 = Zeta(6,n).

1, 65, 47449, 3037465, 47463376609, 47464376609, 5584183099672241, 357389058474664049, 260537105518334091721, 52107472322919827957, 92311616995117182948130877, 92311647383100199924330877, 445570781131605573859221176881493, 445570839299219762020391212081493

Offset: 1

Wolfdieter Lang, Feb 15 2005

For the rationals Zeta(k,n) for k = 1..10 and n = 1..20, see the W. Lang link.

a(n) gives the partial sum, Zeta(6,n), of Euler's (later Riemann's) Zeta(6). Zeta(k,n), k >= 2, is sometimes also called H(k,n) because for k = 1 these would be the harmonic numbers A001008/A002805. However, H(1,n) does not give partial sums of a convergent series.

			The first few fractions are 1, 65/64, 47449/46656, 3037465/2985984, 47463376609/46656000000, ... = A103345/A103346. - _Petros Hadjicostas_, May 10 2020

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Wolfdieter Lang, Rational Zeta(k,n) and more.

Cf. A013664, A291456. For the denominators, see A103346.

For k=1..5, see: A001008/A002805, A007406/A007407, A007408/A007409, A007410/A007480, A099828/A069052.

s=0; lst={}; Do[s+=n^1/n^7; AppendTo[lst,Numerator[s]],{n,3*4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 24 2009 *)
Table[ HarmonicNumber[n, 6] // Numerator, {n, 1, 12}] (* Jean-François Alcover, Dec 04 2013 *)

a(n) = numerator(Sum_{k=1..n} 1/k^6) = numerator(A291456(n)/(n!)^6).
G.f. for rationals Zeta(6, n): polylogarithm(6, x)/(1-x).
Zeta(6, n) = (1/945)*Pi^6 - psi(5, n+1)/5!, see eq. 6.4.3 with m = 5, p. 260, of the Abramowitz-Stegun reference. - Wolfdieter Lang, Dec 03 2013

A291456 a(n) = (n!)^6 * Sum_{i=1..n} 1/i^6.

Original entry on oeis.org

0, 1, 65, 47449, 194397760, 3037656102976, 141727869124448256, 16674281388691716870144, 4371079210518164503303028736, 2322975003299339366419974718488576, 2322977286679362958150790503464960000000

Offset: 0

Seiichi Manyama, Aug 24 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..104

Column k=6 of A291556.
Cf. A000254 (k=1), A001819 (k=2), A066989 (k=3), A203229 (k=4), A099827 (k=5).
Cf. A008516, A103345, A103346.

Table[(n!)^6 * Sum[1/i^6, {i, 1, n}], {n, 0, 15}] (* Vaclav Kotesovec, Aug 27 2017 *)

a(0) = 0, a(1) = 1, a(n+1) = (n^6 + (n+1)^6)*a(n) - n^12*a(n-1) for n > 0.
a(n) ~ 8 * Pi^9 * n^(6*n+3) / (945 * exp(6*n)). - Vaclav Kotesovec, Aug 27 2017
a(n) = (n!)^6 * A103345(n)/A103346(n). - Petros Hadjicostas, May 10 2020
Sum_{n>=0} a(n) * x^n / (n!)^6 = polylog(6,x) / (1 - x). - Ilya Gutkovskiy, Jul 15 2020

A103347 Numerators of Sum_{k=1..n} 1/k^7 = Zeta(7,n).

Original entry on oeis.org

1, 129, 282251, 36130315, 2822716691183, 940908897061, 774879868932307123, 99184670126682733619, 650750755630450535274259, 650750820166709327386387, 12681293156341501091194786541177, 12681293507322704937269896541177

Offset: 1

Wolfdieter Lang, Feb 15 2005

a(n) gives the partial sums, Zeta(7,n), of Euler's Zeta(7). 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 A103348 and for the rationals Zeta(7,n) see the W. Lang link under A103345.

Robert Israel, Table of n, a(n) for n = 1..336

For k=1..6 see: A001008/A002805, A007406/A007407, A007408/A007409, A007410/A007480, A099828/A069052, A103345/A103346.

f:= n -> numer(Psi(6,n+1)/720 + Zeta(7)):
map(f, [$1..20]); # Robert Israel, Mar 28 2018

s=0;lst={};Do[s+=n^1/n^8;AppendTo[lst,Numerator[s]],{n,3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 24 2009 *)
Table[ HarmonicNumber[n, 7] // Numerator, {n, 1, 12}] (* Jean-François Alcover, Dec 04 2013 *)

a(n) = numerator(sum_{k=1..n} 1/k^7).

G.f. for rationals Zeta(7, n): polylogarithm(7, x)/(1-x).

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

Original entry on oeis.org

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).

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).

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).

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

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 A000012, A002805, A007407, A007409, A007480, A069052, A103346, A103348, A103350, A103352, A103717.
Numerators are in A322265.
Cf. A103438, A276487.

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

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

A103345 Numerator of Sum_{k=1..n} 1/k^6 = Zeta(6,n).

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A291456 a(n) = (n!)^6 * Sum_{i=1..n} 1/i^6.

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A103347 Numerators of Sum_{k=1..n} 1/k^7 = Zeta(7,n).

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

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

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

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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.

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