A103345 -id:A103345 - cp's OEIS frontend

A007406 Wolstenholme numbers: numerator of Sum_{k=1..n} 1/k^2.

1, 5, 49, 205, 5269, 5369, 266681, 1077749, 9778141, 1968329, 239437889, 240505109, 40799043101, 40931552621, 205234915681, 822968714749, 238357395880861, 238820721143261, 86364397717734821, 17299975731542641, 353562301485889, 354019312583809, 187497409728228241

Offset: 1

N. J. A. Sloane, Mira Bernstein

By Wolstenholme's theorem, p divides a(p-1) for prime p > 3. - T. D. Noe, Sep 05 2002
Also p divides a( (p-1)/2 ) for prime p > 3. - Alexander Adamchuk, Jun 07 2006
The rationals a(n)/A007407(n) converge to Zeta(2) = (Pi^2)/6 = 1.6449340668... (see the decimal expansion A013661).
For the rationals a(n)/A007407(n), n >= 1, see the W. Lang link under A103345 (case k=2).
See the Wolfdieter Lang link under A103345 on Zeta(k, n) with the rationals for k=1..10, g.f.s and polygamma formulas. - Wolfdieter Lang, Dec 03 2013
Denominator of the harmonic mean of the first n squares. - Colin Barker, Nov 13 2014
Conjecture: for n > 3, gcd(n, a(n-1)) = A089026(n). Checked up to n = 10^5. - Amiram Eldar and Thomas Ordowski, Jul 28 2019
True if n is prime, by Wolstenholme's theorem. It remains to show that gcd(n, a(n-1)) = 1 if n > 3 is composite. - Jonathan Sondow, Jul 29 2019
From Peter Bala, Feb 16 2022: (Start)
Sum_{k = 1..n} 1/k^2 = 1 + (1 - 1/2^2)*(n-1)/(n+1) - (1/2^2 - 1/3^2)*(n-1)*(n-2)/((n+1)*(n+2)) + (1/3^2 - 1/4^2)*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) - (1/4^2 - 1/5^2)*(n-1)*(n-2)*(n-3)*(n-4)/((n+1)*(n+2)*(n+3)*(n+4)) + .... Cf. A082687 and A120778.
This identity allows us to extend the definition of Sum_{k = 1..n} 1/k^2 to non-integral values of n. (End)
Numerators of the Eulerian numbers T(-2,k) for k = 0,1..., if T(n,k) is extended to negative n by the recurrence T(n,k) = (k+1)*T(n-1,k) + (n-k)*T(n-1,k-1) (indexed as in A173018). - Michael J. Collins, Oct 10 2024

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 1..1152 (terms 1..200 from T. D. Noe)
Stephen Crowley, Two New Zeta Constants: Fractal String, Continued Fraction, and Hypergeometric Aspects of the Riemann Zeta Function, arXiv:1207.1126 [math.NT], 2012.
Romeo Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
D. Y. Savio, E. A. Lamagna and S.-M. Liu, Summation of harmonic numbers, pp. 12-20 of E. Kaltofen and S. M. Watt, editors, Computers and Mathematics, Springer-Verlag, NY, 1989.
M. D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications , J. Int. Seq. 13 (2010), 10.6.7, Section 4.3.2.
Eric Weisstein's World of Mathematics, Wolstenholme's Theorem
Eric Weisstein's World of Mathematics, Wolstenholme Number

Cf. A001008, A007407 (denominators), A000290, A082687, A120778.

Numbers n such that a(n) is prime are listed in A111354. Primes in {a(n)} are listed in A123751. - Alexander Adamchuk, Oct 11 2006

import Data.Ratio ((%), numerator)
a007406 n = a007406_list !! (n-1)
a007406_list = map numerator $ scanl1 (+) $ map (1 %) $ tail a000290_list
-- Reinhard Zumkeller, Jul 06 2012

[Numerator(&+[1/k^2:k in [1..n]]):n in [1..23]]; // Marius A. Burtea, Aug 02 2019

a:= n-> numer(add(1/i^2, i=1..n)): seq(a(n), n=1..24);  # Zerinvary Lajos, Mar 28 2007

a[n_] := If[ n<1, 0, Numerator[HarmonicNumber[n, 2]]]; Table[a[n], {n, 100}]
Numerator[HarmonicNumber[Range[20],2]] (* Harvey P. Dale, Jul 06 2014 *)

{a(n) = if( n<1, 0, numerator( sum( k=1, n, 1 / k^2 ) ) )} /* Michael Somos, Jan 16 2011 */

Sum_{k=1..n} 1/k^2 = sqrt(Sum_{j=1..n} Sum_{i=1..n} 1/(i*j)^2). - Alexander Adamchuk, Oct 26 2004
G.f. for rationals a(n)/A007407(n), n >= 1: polylog(2,x)/(1-x).
a(n) = Numerator of (Pi^2)/6 - Zeta(2,n). - Artur Jasinski, Mar 03 2010

A007408 Wolstenholme numbers: numerator of Sum_{k=1..n} 1/k^3.

Original entry on oeis.org

1, 9, 251, 2035, 256103, 28567, 9822481, 78708473, 19148110939, 19164113947, 25523438671457, 25535765062457, 56123375845866029, 56140429821090029, 56154295334575853, 449325761325072949, 2207911834254200646437, 245358578943756786493

Offset: 1

N. J. A. Sloane, Mira Bernstein

By Theorem 131 in Hardy and Wright, p^2 divides a(p - 1) for prime p > 5. - T. D. Noe, Sep 05 2002
p^3 divides a(p - 1) for prime p = 37. Primes p such that p divides a((p + 1)/2) are listed in A124787(n) = {3, 11, 17, 89}. - Alexander Adamchuk, Nov 07 2006
a(n)/A007409(n) is the partial sum towards zeta(3), where zeta(s) is the Riemann zeta function. - Alonso del Arte, Dec 30 2012
See the Wolfdieter Lang link under A103345 on Zeta(k, n) with the rationals for k=1..10, g.f.s and polygamma formulas. - Wolfdieter Lang, Dec 03 2013
Denominator of the harmonic mean of the first n cubes. - Colin Barker, Nov 13 2014

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oxford Univ. Press, 1971, page 104.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..200
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
D. Y. Savio, E. A. Lamagna and S.-M. Liu, Summation of harmonic numbers, pp. 12-20 of E. Kaltofen and S. M. Watt, editors, Computers and Mathematics, Springer-Verlag, NY, 1989.
M. D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications , J. Int. Seq. 13 (2010), 10.6.7, Section 4.3.2.

Cf. A001008, A007406, A007409 (denom), A002117, A124787, A249950.

A007408:=n->numer(sum(1/k^3,k=1..n)); map(%,[$1..20]); # M. F. Hasler, Nov 10 2006

Table[Numerator[Sum[1/k^3, {k, n}]], {n, 10}] (* Alonso del Arte, Dec 30 2012 *)
Table[Denominator[HarmonicMean[Range[n]^3]],{n,20}] (* Harvey P. Dale, Aug 20 2017 *)
Accumulate[1/Range[20]^3]//Numerator (* Harvey P. Dale, Aug 28 2023 *)

a(n)=numerator(sum(k=1,n,1/k^3)) \\ Charles R Greathouse IV, Jul 19 2011

from fractions import Fraction
from itertools import accumulate, count, islice
def A007408gen(): yield from map(lambda x: x.numerator, accumulate(Fraction(1, k**3) for k in count(1)))
print(list(islice(A007408gen(), 20))) # Michael S. Branicky, Jun 26 2022

Sum_{k = 1 .. n} 1/k^3 = sqrt(sum_{j = 1 .. n} sum_{i = 1 .. n} 1/(i * j)^3). - Alexander Adamchuk, Oct 26 2004

A007410 Numerator of Sum_{k=1..4} k^(-4).

Original entry on oeis.org

1, 17, 1393, 22369, 14001361, 14011361, 33654237761, 538589354801, 43631884298881, 43635917056897, 638913789210188977, 638942263173398977, 18249420414596570742097, 18249859383918836502097, 18250192489014819937873

Offset: 1

N. J. A. Sloane, Mira Bernstein

p divides a(p-1) for prime p > 5. p divides a((p-1)/2) for prime p > 5. p^2 divides a((p-1)/2) for prime p = 31, 37. - Alexander Adamchuk, Jul 07 2006
p^2 divides a(p-1) for prime p = 37. - Alexander Adamchuk, Nov 07 2006
Denominators are A007480. See the W. Lang link under A103345 for the rationals and more.
The limit of the rationals Zeta(n) := Sum_{k=1..n} 1/k^4 as n -> infinity is (Pi^4)/90, which is approximately 1.082323234. See A013662.

D. Y. Savio, E. A. Lamagna, and S.-M. Liu, Summation of harmonic numbers, pp. 12-20, in: E. Kaltofen and S. M. Watt, editors, Computers and Mathematics, Springer-Verlag, NY, 1989.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..200
Hisanori Mishima, Factorizations of many number sequences.
Hisanori Mishima, Factorizations of many number sequences.

Cf. A001008, A007406, A007408, A007480, A013662.

Numerator[Table[Sum[1/k^4,{k,1,n}],{n,1,20}]] (* Alexander Adamchuk, Jul 07 2006 *)
Accumulate[1/Range[20]^4]//Numerator (* Harvey P. Dale, Jun 28 2020 *)

a(n)=numerator(sum(k=1,n,1/k^4)) \\ Charles R Greathouse IV, Jul 19 2011

Numerators of the coefficients in the expansion of PolyLog(4, x)/(1 - x). - Ilya Gutkovskiy, Apr 10 2017

A099828 Numerator of the generalized harmonic number H(n,5) = Sum_{k=1..n} 1/k^5.

Original entry on oeis.org

1, 33, 8051, 257875, 806108207, 268736069, 4516906311683, 144545256245731, 105375212839937899, 105376229094957931, 16971048697474072945481, 16971114472329088045481, 6301272372663207205033976933

Offset: 1

Alexander Adamchuk, Oct 27 2004

From Alexander Adamchuk, Nov 07 2006: (Start)
a(n) is prime for n = {23, 25, 85, 147, 167, ...}.
There is a Wolstenholme-like theorem: p divides a(p-1) for prime p and p^2 divides a(p-1) for prime p > 7.
Also, p^3 divides a(p-1) for prime p = 5; p divides a((p-1)/2) for prime p = 37; p divides a((p-1)/3) for prime p = 37; p divides a((p-1)/4) for prime p = 37; p divides a((p-1)/5) for prime p = 11; p^2 divides a((p-1)/6) for prime p = 37; p divides a((p+1)/4) for prime p = 83; p divides a((p+1)/5) for prime p = 29; and p divides a((p+1)/6) for prime p = 11. (End)
See the Wolfdieter Lang link for information about Zeta(k, n) = H(n, k) with the rationals for k = 1..10, g.f.s, and polygamma formulas. - Wolfdieter Lang, Dec 03 2013

			H(n,5) = {1, 33/32, 8051/7776, 257875/248832, ... } = A099828/A069052.
For example, a(2) = numerator(1 + 1/2^5) = numerator(33/32) = 33 and a(3) = numerator(1 + 1/2^5 + 1/3^5) = numerator(8051/7776) = 8051. [Edited by _Petros Hadjicostas_, May 10 2020]

Alexander Adamchuk, Nov 07 2006, Table of n, a(n) for n = 1..100
Wolfdieter Lang, Rational Zeta(k,n) and more.
Eric Weisstein's World of Mathematics, Wolstenholme's Theorem.

Denominators are A069052.
A099827 = H(n,5) multiplied by (n!)^5.
Cf. A001008, A007406, A007408, A007410.

Numerator[Table[Sum[1/k^5, {k, 1, n}], {n, 1, 20}]]
Numerator[Table[HarmonicNumber[n, 5], {n, 1, 20}]]
Table[Numerator[Sum[1/k^5,{k,1,n}]],{n,1,100}] (* Alexander Adamchuk, Nov 07 2006 *)

a(n) = numerator(sum(k=1, n, 1/k^5)); \\ Michel Marcus, May 10 2020

a(n) = numerator(Sum_{k=1..n} 1/k^5) = numerator(HarmonicNumber[n, 5]).

A136675 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^3.

Original entry on oeis.org

1, 7, 197, 1549, 195353, 194353, 66879079, 533875007, 14436577189, 14420574181, 19209787242911, 19197460851911, 42198121495296467, 6025866788581781, 6027847576222613, 48209723660000029, 236907853607882606477

Offset: 1

Alexander Adamchuk, Jan 16 2008

a(n) is prime for n in A136683.

Lim_{n -> infinity} a(n)/A334582(n) = A197070. - Petros Hadjicostas, May 07 2020

			The first few fractions are 1, 7/8, 197/216, 1549/1728, 195353/216000, 194353/216000, 66879079/74088000, 533875007/592704000, ... = a(n)/A334582(n). - _Petros Hadjicostas_, May 06 2020

Robert Israel, Table of n, a(n) for n = 1..768
Eric Weisstein's World of Mathematics, Harmonic Number.
Wikipedia, Dirichlet eta function.

Cf. A001008, A007406, A007408, A007410, A058313, A099828, A103345, A119682, A120296, A136676, A136677, A136681, A136682, A136683, A136684, A136685, A136686, A197070, A334582 (denominators).

map(numer,ListTools:-PartialSums([seq((-1)^(k+1)/k^3, k=1..100)])); # Robert Israel, Nov 09 2023

(* Program #1 *) Table[Numerator[Sum[(-1)^(k+1)/k^3, {k,1,n}]], {n,1,50}]
(* Program #2 *) Numerator[Accumulate[Table[(-1)^(k+1) 1/k^3, {k,50}]]] (* Harvey P. Dale, Feb 12 2013 *)

a(n) = numerator(sum(k=1, n, (-1)^(k+1)/k^3)); \\ Michel Marcus, May 07 2020

A136677 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^6.

Original entry on oeis.org

1, 63, 45991, 2942695, 45982595359, 5109066151, 601081707598999, 38469080386820311, 252396118308232060471, 252395862211967012407, 447134922152359540530757327, 447134770212444455649757327, 2158234586764514215343657417779543, 308319185132349039219686748825649

Offset: 1

Alexander Adamchuk, Jan 16 2008

p divides a(p-1) for prime p > 2. a(n) is prime for n = {19, 47, 164, ...} = A136686.

Lim_{n -> infinity} a(n)/A334605(n) = A275703 = (31/32)*A013664. - Petros Hadjicostas, May 07 2020

			The first few fractions are 1, 63/64, 45991/46656, 2942695/2985984, 45982595359/46656000000, 5109066151/5184000000, ... = a(n)/A334605(n). - _Petros Hadjicostas_, May 07 2020

Harvey P. Dale, Table of n, a(n) for n = 1..382
Eric Weisstein's World of Mathematics, Harmonic Number.

Cf. A013664, A058313, A119682, A120296, A136675, A136676.
Cf. A001008, A007406, A007408, A007410, A099828, A103345, A275703.
Cf. A136681, A136682, A136683, A136684, A136685, A136686, A334605 (denominators).

Table[ Numerator[ Sum[ (-1)^(k+1)/k^6, {k,1,n} ] ], {n,1,30} ]
Accumulate[Table[(-1)^(k+1)/k^6,{k,20}]]//Numerator (* Harvey P. Dale, Aug 21 2023 *)

A136676 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^5.

Original entry on oeis.org

1, 31, 7565, 241837, 755989457, 755889457, 12705011703799, 406547611705943, 98792790681344149, 98791774426324117, 15910615688635928566967, 15910549913780913466967, 5907492176026179821253778331

Offset: 1

Alexander Adamchuk, Jan 16 2008

a(n) is prime for n in A136685.

Lim_{n -> infinity} a(n)/A334604(n) = A267316 = (15/16)*A013663. - Petros Hadjicostas, May 07 2020

			The first few fractions are 1, 31/32, 7565/7776, 241837/248832, 755989457/777600000, 755889457/777600000, ... = a(n)/A334604(n). - _Petros Hadjicostas_, May 07 2020

Eric Weisstein's World of Mathematics, Harmonic Number.

Cf. A001008, A007406, A007408, A007410, A013663, A058313, A099828, A103345, A119682, A120296, A136675, A136677, A136681, A136682, A136683, A136684, A136685, A136686, A267316, A334604 (denominators).

Table[ Numerator[ Sum[ (-1)^(k+1)/k^5, {k,1,n} ] ], {n,1,30} ]

a(n) = numerator(sum(k=1, n, (-1)^(k+1)/k^5)); \\ Michel Marcus, May 07 2020

A069052 Denominator of Sum_{i = 1..n} 1/i^5.

Original entry on oeis.org

1, 32, 7776, 248832, 777600000, 259200000, 4356374400000, 139403980800000, 101625502003200000, 101625502003200000, 16366888723117363200000, 16366888723117363200000, 6076911214672415134617600000

Offset: 1

Benoit Cloitre, Apr 03 2002

If 1 <= n <= 19, a(n) = A007480(n) * A002805(n) = denominator(Sum_{i = 1..n} 1/i^4) * denominator(Sum_{i = 1..n} 1/i).

			The first few fractions are 1, 33/32, 8051/7776, 257875/248832, ... = A099828/A069052. - _Petros Hadjicostas_, May 10 2020

Wolfdieter Lang, Rational Zeta(k,n) and more.

Numerators are A099828.

Cf. A002805, A007480, A099827.

s = 0; lst = {}; Do[s += 1/n^5; AppendTo[lst, Denominator[s]], {n, 3 * 4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 24 2009 *)
Denominator[Table[Sum[1/i^5, {i, n}], {n, 20}]] (* Alonso del Arte, May 10 2020 *)

a(n) = denominator(Sum_{k=1..n} 1/k^5) = denominator(A099827(n)/(n!)^5). - Petros Hadjicostas, May 10 2020

A103346 Denominators of Sum_{k=1..n} 1/k^6 = Zeta(6,n).

Original entry on oeis.org

1, 64, 46656, 2985984, 46656000000, 46656000000, 5489031744000000, 351298031616000000, 256096265048064000000, 51219253009612800000, 90738031080962661580800000, 90738031080962661580800000

Offset: 1

Wolfdieter Lang, Feb 15 2005

For the numerators and comments, see A103345.

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

Cf. A103345, A291456.

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

a(n) = denominator(Sum_{k=1..n} 1/k^6) = denominator(A291456(n)/(n!)^6). - Petros Hadjicostas, May 10 2020

A136681 Numbers k such that A058313(k) is prime.

Original entry on oeis.org

3, 4, 5, 6, 9, 10, 13, 16, 17, 18, 37, 43, 58, 121, 124, 126, 137, 203, 247, 283, 285, 286, 289, 317, 424, 508, 751, 790, 937, 958, 1066, 1097, 1151, 1166, 1194, 1199, 1235, 1414, 1418, 1460, 1498, 1573, 2090, 2122, 2691, 2718, 3030, 3426, 3600, 3653, 3737

Offset: 1

Alexander Adamchuk, Jan 16 2008

nonn

A058313(k) = Numerator of Sum_{j=1..k} (-1)^(j+1)/j.

Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A058313, A119682, A120296, A136675, A136676, A136677.
Cf. A001008, A007406, A007408, A007410, A099828, A103345.
Cf. A136682, A136683, A136684, A136685, A136686.

Do[ f=Numerator[ Sum[ (-1)^(k+1)*1/k, {k,1,n} ] ]; If[ PrimeQ[f], Print[ {n,f} ] ], {n,1,317} ]

isok(n) = isprime(numerator(sum(k=1, n, (-1)^(k+1)/k))); \\ Michel Marcus, Mar 14 2019

a(25)-a(30) from James R. Buddenhagen, Sep 22 2015

a(31)-a(51) from Amiram Eldar, Mar 14 2019

cp's OEIS Frontend

A007406 Wolstenholme numbers: numerator of Sum_{k=1..n} 1/k^2.

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A007408 Wolstenholme numbers: numerator of Sum_{k=1..n} 1/k^3.

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A007410 Numerator of Sum_{k=1..4} k^(-4).

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A099828 Numerator of the generalized harmonic number H(n,5) = Sum_{k=1..n} 1/k^5.

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A136675 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^3.

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A136677 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^6.

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A136676 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^5.

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