A007406 -id:A007406 - cp's OEIS frontend

A373439 Numerator of sum of reciprocals of square divisors of n.

1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 5, 1, 1, 1, 21, 1, 10, 1, 5, 1, 1, 1, 5, 26, 1, 10, 5, 1, 1, 1, 21, 1, 1, 1, 25, 1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 21, 50, 26, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 1, 1, 10, 85, 1, 1, 1, 5, 1, 1, 1, 25, 1, 1, 26, 5, 1, 1, 1, 21, 91, 1, 1, 5, 1

Offset: 1

Ilya Gutkovskiy, Jun 05 2024

			1, 1, 1, 5/4, 1, 1, 1, 5/4, 10/9, 1, 1, 5/4, 1, 1, 1, 21/16, 1, 10/9, 1, 5/4, 1, 1, 1, 5/4, 26/25, ...

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

Cf. A007406, A017665, A017667, A028235, A035316, A332880, A373440 (denominators).

nmax = 85; CoefficientList[Series[Sum[x^(k^2)/(k^2 (1 - x^(k^2))), {k, 1, nmax}], {x, 0, nmax}], x] // Rest // Numerator
f[p_, e_] := (p^2 - p^(-2*Floor[e/2]))/(p^2-1); a[1] = 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Jun 26 2024 *)

a(n) = numerator(sumdiv(n, d, if (issquare(d), 1/d))); \\ Michel Marcus, Jun 05 2024

Numerators of coefficients in expansion of Sum_{k>=1} x^(k^2)/(k^2*(1 - x^(k^2))).
a(n) is the numerator of Sum_{d^2|n} 1/d^2.
From Amiram Eldar, Jun 26 2024: (Start)
Let f(n) = a(n)/A373440(n). Then:
f(n) is multiplicative with f(p^e) = (p^2 - p^(-2*floor(e/2)))/(p^2-1).
Dirichlet g.f. of f(n): zeta(s) * zeta(2*s+2).
Sum_{k=1..n} f(k) ~ zeta(4) * n. (End)

A070985 Number of terms in the simple continued fraction for Sum_{k=1..n} 1/k^2.

Original entry on oeis.org

1, 2, 5, 7, 9, 7, 10, 20, 18, 14, 22, 19, 18, 24, 26, 24, 30, 30, 28, 37, 25, 30, 35, 35, 34, 38, 47, 52, 49, 54, 40, 49, 49, 69, 57, 67, 78, 67, 67, 68, 67, 64, 65, 86, 76, 81, 92, 79, 83, 83, 95, 82, 85, 80, 84, 95, 92, 91, 121, 105, 100, 108, 111, 109, 118, 105, 110, 88

Offset: 1

Benoit Cloitre, May 18 2002

Sum_{k>=1} 1/k^2 = zeta(2) = Pi^2/6.

			The simple continued fraction for Sum_{k=1..10} 1/k^2 is [1, 1, 1, 4, 1, 1, 10, 4, 1, 2, 5, 2, 1, 24] which contains 14 terms, hence a(10) = 14.

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

Cf. A013679, A055573, A007406, A007407.

lcf[f_] := Length[ContinuedFraction[f]]; lcf /@ Accumulate[Table[1/k^2, {k, 1, 100}]] (* Amiram Eldar, Apr 30 2022 *)

for(n=1,100,print1(length(contfrac(sum(i=1,n,1/i^2))),","))

Limit_{n ->infinity} a(n)/n = C =1.6....

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).

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

Original entry on oeis.org

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}]].

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.

A246498 Numerator of the harmonic mean of the first n squares.

Original entry on oeis.org

1, 8, 108, 576, 18000, 21600, 1234800, 5644800, 57153600, 12700800, 1690476480, 1844156160, 337634256960, 363606122880, 1947889944000, 8310997094400, 2551995545299200, 2702112930316800, 1029655143835718400, 216769503965414400, 4645060799258880

Offset: 1

Colin Barker, Nov 14 2014

nonn

			a(3) = 108 because the first 3 squares are [1,4,9] and 3 / (1/1+1/4+1/9) = 108/49.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A007406 (denominators).

A246498 := proc(n::integer)
    n/add(1/k^2,k=1..n) ;
    numer(%) ;
end proc:
seq(A246498(n),n=1..13) ; # R. J. Mathar, Nov 05 2019

Table[Numerator[n/Sum[1/k^2,{k,1,n}]],{n,1,20}] (* Vaclav Kotesovec, Nov 14 2014 *)

harmonicmean(v) = #v / sum(k=1, #v, 1/v[k])
s=vector(30); for(k=1, #s, s[k]=numerator(harmonicmean(vector(k, i, i^2)))); s

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

A309391 a(n) = gcd(n, A064169(n-2)) for n > 2.

Original entry on oeis.org

3, 1, 5, 1, 7, 1, 1, 1, 11, 1, 13, 1, 1, 1, 17, 1, 19, 1, 1, 1, 23, 1, 5, 1, 1, 1, 29, 1, 31, 1, 1, 1, 1, 1, 37, 1, 1, 1, 41, 1, 43, 1, 1, 1, 47, 1, 7, 1, 1, 1, 53, 1, 1, 1, 1, 1, 59, 1, 61, 1, 1, 1, 1, 1, 67, 1, 1, 1, 71, 1, 73, 1, 1, 1, 1, 1, 79, 1, 1, 1, 83, 1, 1, 1, 1, 11, 89, 1

Offset: 3

Amiram Eldar and Thomas Ordowski, Jul 28 2019

nonn

Probably, there are no composite terms in this sequence.
For n > 2, a(n) = gcd(n, A001008(n-1)).
By Wolstenholme's theorem, if p is an odd prime, then a(p) = p.
Conjecture: for n > 2, if a(n) = n, then n is a prime.
If so, then there are no pseudoprimes n such that a(n) = n.
Composite numbers m <> p^2 for which a(m) > 1 are 88, 1290, 9339, ...

			a(25) = gcd(25, A064169(25-2)) = gcd(25, 325333835) = 5,
a(25) = gcd(25, A001008(25-1)) = gcd(25, 1347822955) = 5.
It should be noted that a(88) = 11, a(1290) = 43, a(9339) = 11, ...

Robert Israel, Table of n, a(n) for n = 3..10000
Romeo Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012), arXiv:1111.3057 [math.NT], 2001.
Eric Weisstein's World of Mathematics, Wolstenholme's Theorem.
Wikipedia, Wolstenholme's theorem.

Cf. A001008, A002805, A007406 (see our comment), A064169, A065091, A089026, A309397.

[Gcd(k, Numerator(a)-Denominator(a)) where a is HarmonicNumber(k-2):k in [3..90]]; // Marius A. Burtea, Jul 29 2019

H:= 0:
for n from 3 to 100 do
  H:= H + 1/(n-2);
  A[n]:= igcd(n, numer(H)-denom(H));
od:
seq(A[i],i=3..100); # Robert Israel, Aug 04 2019

a[n_] := GCD[n, Numerator[(h = HarmonicNumber[n-2])] - Denominator[h]]; Array[a, 81, 3]

a(p) = p for every odd prime p.
a(p^2) = p iff p > 3 is a prime.
Note that a(n) >= A089026(n) for n > 2.

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

cp's OEIS Frontend

A373439 Numerator of sum of reciprocals of square divisors of n.

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A070985 Number of terms in the simple continued fraction for Sum_{k=1..n} 1/k^2.

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

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

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A246498 Numerator of the harmonic mean of the first n squares.

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

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