A007407 -id:A007407 - cp's OEIS frontend

A111354 Numbers n such that the numerator of Sum_{i=1..n} (1/i^2), in reduced form, is prime.

2, 7, 13, 19, 121, 188, 252, 368, 605, 745, 1085, 1127, 1406, 1743, 1774, 2042, 2087, 2936, 3196, 3207, 3457, 4045, 7584, 10307, 12603, 12632, 14438, 14526, 14641, 15662, 15950, 16261, 18084, 18937, 19676, 40984, 45531, 46009, 48292, 48590

Offset: 1

Ryan Propper, Nov 05 2005

nonn

Numbers n such that A007406(n) is prime.
Some of the larger entries may only correspond to probable primes.
A007406(n) are the Wolstenholme numbers: numerator of Sum 1/k^2, k = 1..n. Primes in A007406(n) are listed in A123751(n) = A007406(a(n)) = {5,266681,40799043101,86364397717734821,...}.
For prime p>3, Wolstenholme's theorem says that p divides A007406(p-1). Hence n+1 cannot be prime for any n>2 in this sequence. - 12 more terms from T. D. Noe, Nov 11 2005
No other n<50000. All n<=1406 yield provable primes. - T. D. Noe, Mar 08 2006

			A007406(n) begins {1, 5, 49, 205, 5269, 5369, 266681, 1077749, 9778141,...}.
Thus a(1) = 2 because A007406(2) = 5 is prime but A007406(1) = 1 is not prime.
a(2) = 7 because A007406(7) = 266681 is prime but all A007406(k) are composite for 2 < k < 7.

Carlos M. da Fonseca, M. Lawrence Glasser, Victor Kowalenko, Generalized cosecant numbers and trigonometric inverse power sums, Applicable Analysis and Discrete Mathematics, Vol. 12, No. 1 (2018), 70-109.
Eric Weisstein's World of Mathematics, Wolstenholme's Theorem
Eric Weisstein's World of Mathematics, Harmonic Number.
Eric Weisstein's World of Mathematics, Wolstenholme Number

Cf. A007406 (numerator of Sum_{i=1..n} (1/i^2)).

Cf. A123751, A001008, A007407, A067567, A056903.

s = 0; Do[s += 1/n^2; If[PrimeQ[Numerator[s]], Print[n]], {n, 1, 10^4}]
Module[{nn=10400,t},t=Accumulate[1/Range[nn]^2];Select[Thread[{Range[nn],Numerator[t]}],PrimeQ[#[[2]]]&]][[;;,1]] (* The program generates the first 24 terms of the sequence. *) (* Harvey P. Dale, May 18 2025 *)

12 more terms from T. D. Noe, Nov 11 2005
More terms from T. D. Noe, Mar 08 2006
Additional comments from Alexander Adamchuk, Oct 11 2006
Edited by N. J. A. Sloane, Nov 11 2006

A120286 Numerator of 1/n^2 + 2/(n-1)^2 + 3/(n-2)^2 +...+ (n-1)/2^2 + n.

Original entry on oeis.org

1, 9, 65, 725, 3899, 28763, 419017, 864669, 7981633, 3586319, 200763407, 2649665993, 34899471137, 176508049513, 356606957297, 12234391348253, 209672027529221, 4012917216669239, 15350275129353301, 15443118015171841

Offset: 1

Alexander Adamchuk, Jul 07 2006

p^2 divides a(p-1) for prime p>2. p divides a(p-2) for prime p>3.

Cf. A027612, A001008, A007406, A007407, A370774 (denom.).

Numerator[Table[Sum[Sum[1/i^2,{i,1,k}],{k,1,n}],{n,1,30}]]
Table[-EulerGamma + HarmonicNumber[1 + n, 2] + n*HarmonicNumber[1 + n, 2] - PolyGamma[0, 2 + n], {n, 1, 20}] // Numerator (* Vaclav Kotesovec, May 02 2024 *)

from fractions import Fraction
def A120286(n): return sum(Fraction(n-i+1,i**2) for i in range(1,n+1)).numerator # Chai Wah Wu, May 01 2024

a(n) = numerator[Sum[Sum[1/i^2,{i,1,k}],{k,1,n}]].

A123751 Primes in A007406.

Original entry on oeis.org

5, 266681, 40799043101, 86364397717734821, 36190908596780862323291147613117849902036356128879432564211412588793094572280300268379995976006474252029, 334279880945246012373031736295774418479420559664800307123320901500922509788908032831003901108510816091067151027837158805812525361841612048446489305085140033

Offset: 1

Alexander Adamchuk, Oct 11 2006

nonn

A007406 lists the Wolstenholme numbers.

Numbers k such that A007406(k) is prime are listed in A111354.

			A007406 begins {1, 5, 49, 205, 5269, 5369, 266681, 1077749, 9778141, ...}.
Thus a(1) = 5 because A007406(2) = 5 is prime but A007406(1) = 1 is not prime.
a(2) = 266681 because A007406(7) = 266681 is prime but all A007406(k) are composite for 2 < k < 7.

Carlos M. da Fonseca, M. Lawrence Glasser, Victor Kowalenko, Generalized cosecant numbers and trigonometric inverse power sums, Applicable Analysis and Discrete Mathematics, Vol. 12, No. 1 (2018), 70-109.
Eric Weisstein's World of Mathematics, Harmonic Number.
Eric Weisstein's World of Mathematics, Wolstenholme's Theorem.
Eric Weisstein's World of Mathematics, Wolstenholme Number

Cf. A111354, A007406, A001008, A007407, A067567, A056903.

Do[f=Numerator[Sum[1/i^2,{i,1,n}]]; If[PrimeQ[f],Print[{n,f}]],{n,1,250}]

a(n) = A007406(A111354(n)).

A163928 Numerators of the higher order exponential integral constants alpha(2,n).

Original entry on oeis.org

0, 1, 21, 1897, 32197, 20881861, 7139587, 17462165587, 283355376967, 69621962857381, 70246946681461, 1036088178214798501, 1042504974775473001, 29931734181763981573561, 4295332813075795410223, 4312254507400142830831

Offset: 1

Johannes W. Meijer & Nico Baken, Aug 13 2009, Aug 17 2009

See A163927 for information about the alpha(k,n) constants.

Apart from a difference of offset, alpha(2,n) appears to be the multiple harmonic (star) sum Sum_{j = 1..n} 1/j^2 Sum_{k = 1..j} 1/k^2, which has the initial values [1, 21/16, 1897/1296, 32197/20736, 20881861/12960000, 7139587/4320000, ...]. - Peter Bala, Jan 31 2019

			alpha(k=2,n=1) = 0, alpha(k=2,2) = 1, alpha(k=2,3) = 21/16, alpha(k=2,4) = 1897/1296.

Cf. A163929 (denominators).
Cf. A163927 (alpha(k,n)) and A090998 (gamma(k,n)).
Cf. A007406, A007407.

nmax:=17; rowk:=2; kmax:=nmax: k:=0: for n from 1 to nmax do alpha(k,n):=1 od: for k from 1 to kmax do for n from 1 to nmax do alpha(k,n) := (1/k)*sum(sum(p^(-2*(k-i)),p=0..n-1)*alpha(i, n),i=0..k-1) od; od: seq(alpha(rowk, n),n=1..nmax);

alpha(k,n) = (1/k)*Sum_{i=0..k-1} (Sum_{p=0..n-1} p^(-2*(k-i))*alpha(i, n) with alpha(0,n) = 1, with k = 2 and n >= 1. alpha(1,n) = A007406(n-1)/A007407(n-1) for n >= 2.

A276485 Numerator of Sum_{k=1..n} 1/k^n.

Original entry on oeis.org

1, 5, 251, 22369, 806108207, 47464376609, 774879868932307123, 248886558707571775009601, 4106541588424891370931874221019, 413520574906423083987893722912609, 7429165883912264897181708263009894640627544300697

Offset: 1

Ilya Gutkovskiy, Sep 05 2016

Also numerators 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) = 251, 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, A276487 (denominators).

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

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

A345651 Fourth column of A008296.

Original entry on oeis.org

1, 10, 25, -35, 49, 0, -820, 9020, -87164, 859144, -8965320, 100136400, -1199838576, 15406135488, -211479420096, 3094582896000, -48129022468224, 793274283938304, -13818265424460288, 253731538514893824, -4899371564756837376, 99261476593521868800

Offset: 4

Luca Onnis, Aug 26 2021

sign

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 139, b(n,4).

Cf. A008296, A045406, A081048.

Cf. A001008, A002805, A007406, A007407.

b:= proc(n, k) option remember; `if`(n=k, 1, `if`(k=0, 0,
      (n-1)*b(n-2, k-1)+b(n-1, k-1)+(k-n+1)*b(n-1, k)))
    end:
a:= n-> b(n, 4):
seq(a(n), n=4..28);  # Alois P. Heinz, Aug 26 2021
# alternative
seq(A008296(n,4),n=4..70) ; # R. J. Mathar, Sep 15 2021

a[1, 1] = a[2, 1] = 1; a[n_, 1] = (-1)^n (n - 2)!;
a[n_, n_] = 1;
a[n_, k_] := a[n, k] = (n - 1) a[n - 2, k - 1] +
    a[n - 1, k - 1] + (k - n + 1) a[n - 1, k];
Flatten[Table[N[a[n + 4, 4], 10], {n, 1, 400}]]

a(n) = sum(m=4, n, binomial(m, 4)*4^(m-4)*stirling(n, m, 1)); \\ Michel Marcus, Sep 14 2021

a(n) = A008296(n,4).
a(n) = (-1)^n*(4*H(n-5,1)^3 + 8*H(n-5,3) - 12*H(n-5,2)*H(n-5,1) - 25*H(n-5,1)^2 + 25*H(n-5,2) + 35*H(n-5,1) - 10)*(n-5)! for n >= 5 where H(n,1) = Sum_{j=1..n} 1/j is the n-th harmonic number, H(n,2) = Sum_{j=1..n} 1/j^2 and H(n,3) = Sum_{j=1..n} 1/j^3.
a(n) = Sum_{m=4..n} binomial(m,4) * 4^(m-4) * Stirling1(n,m). - Alois P. Heinz, Aug 26 2021
Conjecture: D-finite with recurrence a(n) +2*(2*n-13)*a(n-1) +(6*n^2-84*n+295)*a(n-2) +(2*n-15)*(2*n^2-30*n+113)*a(n-3) +(n-8)^4*a(n-4)=0. - R. J. Mathar, Sep 15 2021

A347276 Third column of A008296.

Original entry on oeis.org

1, 6, 5, -15, 49, -196, 944, -5340, 34716, -254760, 2078856, -18620784, 180973584, -1887504768, 20887922304, -242111586816, 2889841121280, -34586897978880, 393722260047360, -3659128846433280, 5687630494110720, 1137542166526464000, -49644151627682304000

Offset: 3

Luca Onnis, Aug 25 2021

sign

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 139, b(n,3).

Cf. A008296, A045406, A081048.
Cf. A001008, A002805, A007406, A007407.
Cf. A048994, A008275.

b:= proc(n, k) option remember; `if`(n=k, 1, `if`(k=0, 0,
      (n-1)*b(n-2, k-1)+b(n-1, k-1)+(k-n+1)*b(n-1, k)))
    end:
a:= n-> b(n, 3):
seq(a(n), n=3..30);  # Alois P. Heinz, Aug 25 2021

a[1, 1] = a[2, 1] = 1; a[n_, 1] = (-1)^n (n - 2)!;
a[n_, n_] = 1;
a[n_, k_] :=  a[n, k] = (n - 1) a[n - 2, k - 1] +
    a[n - 1, k - 1] + (k - n + 1) a[n - 1, k];
Flatten[Table[a[n + 3, 3], {n, 0, 30}]]

a(n) = sum(m=3, n, binomial(m, 3)*3^(m-3)*stirling(n, m, 1)); \\ Michel Marcus, Sep 14 2021

a(n) = A008296(n,3).
a(n) = (-1)^n*(3*H(n-4,1)^2 - 3*H(n-4,2) - 11*H(n-4,1) + 6)*(n-4)! for n >= 4, where H(n,1) = Sum_{j=1..n} 1/j = A001008(n)/A002805(n) is the n-th harmonic number and H(n,2) = Sum_{j=1..n} 1/j^2 = A007406(n)/A007407(n).
a(n) = Sum_{m=3..n} binomial(m,3) * 3^(m-3) * Stirling1(n,m). - Alois P. Heinz, Aug 26 2021

A060944 a(n) = n!^2 * Sum_{k=1..n} Sum_{j=1..k} 1/j^2.

Original entry on oeis.org

1, 9, 130, 2900, 93576, 4141872, 241353792, 17929776384, 1655071418880, 185914776960000, 24978180045312000, 3955930130221056000, 729464836964806656000, 154952762244805582848000, 37566943754471090749440000, 10310706109241121091092480000

Offset: 1

Leroy Quet, May 07 2001

Sum of generalized harmonic numbers squared multiplied by (n!)^2. agenh(n) = Sum_{k=1..n} HarmonicNumber(k, 2), where HarmonicNumber(n, j) = Sum_{k = 1..n} 1/k^j. - Alexander Adamchuk, Oct 27 2004

			a(3) = 6^2 *(1 + (1 + 1/2^2) + (1 + 1/2^2 + 1/3^2)) = 130.

Harry J. Smith, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A001705, A007406, A007407.

[(Factorial(n))^2*(&+[(1+j)/(n-j)^2: j in [0..n-1]]): n in [1..15]]; // G. C. Greubel, Apr 09 2021

A060944:= n-> (n!)^2*add((1+j)/(n-j)^2, j=0..n-1); seq(A060944(n), n=1..15); # G. C. Greubel, Apr 09 2021

Table[(n!)^2*Sum[(k+1)/(n-k)^2, {k, 0, n-1}], {n, 1, 10}]

a(n)={n!^2 * sum(k=1, n, sum(j=1, k, 1/j^2))} \\ Harry J. Smith, Jul 15 2009

[(factorial(n))^2*sum((1+j)/(n-j)^2 for j in (0..n-1)) for n in (1..15)] # G. C. Greubel, Apr 09 2021

From Alexander Adamchuk, Oct 27 2004: (Start)
a(n) = (n!)^2 * Sum_{k=0..n-1} (k+1)/(n-k)^2.
a(n) = (n!)^2 * Sum_{k=1..n} HarmonicNumber(k, 2), where HarmonicNumber(k, 2) = A007406(k) / A007407(k). (End)
Sum_{n>=1} a(n) * x^n / (n!)^2 = polylog(2,x) / (1 - x)^2. - Ilya Gutkovskiy, Jul 15 2020

A068584 Numbers k such that the denominator of (Sum_{j=1..k} 1/j)^2 equals the denominator of Sum_{j=1..k} 1/j^2.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 15, 16, 17, 28, 29, 30, 31, 32, 49, 91, 92, 93, 94, 95, 96, 97, 98, 99, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 243, 244, 245, 246, 247, 248, 249, 968, 969, 970, 971, 972, 973, 974, 975, 976, 977, 978, 979, 980, 981, 982, 983, 984

Offset: 1

Benoit Cloitre, Mar 27 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002805, A007407.

s = {}; sum1 = sum2 = 0; Do[sum1 += 1/j; sum2 += 1/j^2; If[Denominator[sum1^2] == Denominator[sum2], AppendTo[s, j]], {j, 1, 1000}]; s (* Amiram Eldar, Feb 18 2021 *)

isok(k) = denominator(sum(j=1, k, 1/j)^2) == denominator(sum(j=1, k, 1/j^2)); \\ Michel Marcus, Feb 15 2021

Numbers k such that A002805(k)^2 = A007407(k).

A073526 Denominator of Sum_{k=1..n} 1/k^2 is a square.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 16, 17, 18, 19, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 49, 75, 76, 77, 91, 92, 93, 94, 95, 96, 97, 98, 99, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 153, 154, 155, 205, 206, 207, 208, 209, 210, 211

Offset: 1

N. J. A. Sloane, Aug 30 2002

Cf. A007407, A073527.

Select[Range[300],IntegerQ[Sqrt[Denominator[Sum[1/k^2,{k,#}]]]]&] (* Harvey P. Dale, Mar 28 2012 *)

{n: A007407(n) in A000290} . - R. J. Mathar, Oct 03 2014

More terms from Matthew Conroy, Sep 09 2002

cp's OEIS Frontend

A111354 Numbers n such that the numerator of Sum_{i=1..n} (1/i^2), in reduced form, is prime.

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A120286 Numerator of 1/n^2 + 2/(n-1)^2 + 3/(n-2)^2 +...+ (n-1)/2^2 + n.

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A123751 Primes in A007406.

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A163928 Numerators of the higher order exponential integral constants alpha(2,n).

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A276485 Numerator of Sum_{k=1..n} 1/k^n.

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A345651 Fourth column of A008296.

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A347276 Third column of A008296.

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A060944 a(n) = n!^2 * Sum_{k=1..n} Sum_{j=1..k} 1/j^2.

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