A072691 -id:A072691 - cp's OEIS frontend

A258984 Decimal expansion of the multiple zeta value (Euler sum) zetamult(4,2).

0, 8, 8, 4, 8, 3, 3, 8, 2, 4, 5, 4, 3, 6, 8, 7, 1, 4, 2, 9, 4, 3, 2, 7, 8, 3, 9, 0, 8, 5, 7, 6, 0, 4, 5, 6, 6, 4, 7, 9, 7, 8, 7, 5, 2, 3, 8, 6, 7, 5, 0, 5, 9, 1, 6, 7, 4, 8, 8, 9, 2, 7, 6, 5, 5, 9, 4, 7, 4, 2, 7, 8, 9, 2, 8, 7, 4, 3, 5, 7, 1, 4, 5, 5, 8, 2, 7, 7, 9, 4, 6, 0, 0, 4, 7, 0, 5, 8, 6, 6, 1, 9, 5, 5, 9, 6, 6, 7

Offset: 0

Jean-François Alcover, Jun 16 2015

			0.088483382454368714294327839085760456647978752386750591674889276559474...

Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function

Cf. A072691 (zetamult(1,1)), A197110 (zetamult(2,2)), A258983 (zetamult(3,2)), A258985 (5,2), A258947 (6,2), A258986 (2,3), A258987 (3,3), A258988 (4,3), A258982 (5,3), A258989 (2,4), A258990 (3,4), A258991 (4,4).

Join[{0}, RealDigits[Zeta[3]^2 - (4/3)*Zeta[6], 10, 107] // First]

zetamult([4,2]) \\ Charles R Greathouse IV, Jan 21 2016

zetamult(4,2) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^4*n^2)) = zeta(3)^2 - (4/3)*zeta(6).

A258985 Decimal expansion of the multiple zeta value (Euler sum) zetamult(5,2).

Original entry on oeis.org

0, 3, 8, 5, 7, 5, 1, 2, 4, 3, 4, 2, 7, 5, 3, 2, 5, 5, 5, 0, 5, 9, 2, 5, 4, 6, 4, 3, 7, 2, 9, 9, 5, 5, 7, 0, 0, 1, 9, 7, 3, 4, 8, 4, 1, 6, 9, 8, 9, 0, 9, 0, 0, 8, 3, 3, 1, 0, 4, 9, 3, 7, 2, 9, 3, 3, 5, 8, 2, 3, 6, 5, 9, 1, 0, 8, 4, 5, 3, 8, 3, 6, 5, 5, 6, 8, 4, 8, 8, 2, 9, 4, 6, 4, 5, 6, 4, 7, 3, 1, 5, 5, 6, 4, 9

Offset: 0

Jean-François Alcover, Jun 16 2015

			0.03857512434275325550592546437299557001973484169890900833104937293358...

Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function

Cf. A072691 (zetamult(1,1)), A197110 (zetamult(2,2)), A258983 (zetamult(3,2)), A258984 (4,2), A258947 (6,2), A258986 (2,3), A258987 (3,3), A258988 (4,3), A258982 (5,3), A258989 (2,4), A258990 (3,4), A258991 (4,4).

Join[{0}, RealDigits[5*Zeta[2]*Zeta[5] + 2*Zeta[3]*Zeta[4] - 11*Zeta[7], 10, 104] // First]

zetamult([5,2]) \\ Charles R Greathouse IV, Jan 21 2016

zetamult(5,2) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^5*n^2)) = 5*zeta(2)*zeta(5) + 2*zeta(3)*zeta(4) - 11*zeta(7).

A258986 Decimal expansion of the multiple zeta value (Euler sum) zetamult(2,3).

Original entry on oeis.org

7, 1, 1, 5, 6, 6, 1, 9, 7, 5, 5, 0, 5, 7, 2, 4, 3, 2, 0, 9, 6, 9, 7, 3, 8, 0, 6, 0, 8, 6, 4, 0, 2, 6, 1, 2, 0, 9, 2, 5, 6, 1, 2, 0, 4, 4, 3, 8, 3, 3, 9, 2, 3, 6, 4, 9, 2, 2, 2, 2, 4, 9, 6, 4, 5, 7, 6, 8, 6, 0, 8, 5, 7, 4, 5, 0, 5, 8, 2, 6, 5, 1, 1, 5, 4, 2, 5, 2, 3, 4, 4, 6, 3, 6, 0, 0, 7, 9, 8, 9, 6, 4, 1

Offset: 0

Jean-François Alcover, Jun 16 2015

			0.711566197550572432096973806086402612092561204438339236492222496457686...

Dominique Manchon, Arborified multiple zeta values, arXiv:1603.01498 [math.CO], 2016.
Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function

Cf. A072691 (zetamult(1,1)), A197110 (zetamult(2,2)), A258983 (zetamult(3,2)), A258984 (4,2), A258985 (5,2), A258947 (6,2), A258987 (3,3), A258988 (4,3), A258982 (5,3), A258989 (2,4), A258990 (3,4), A258991 (4,4).

Cf. A013663 (zeta(5)), A183699 (zeta(2)*zeta(3)).

RealDigits[(9/2)*Zeta[5] - 2*Zeta[2]*Zeta[3], 10, 103] // First

zetamult([2,3]) \\ Charles R Greathouse IV, Jan 21 2016

zetamult(2,3) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^2*n^3)) = (9/2)*zeta(5) - 2*zeta(2)*zeta(3).

Equals Sum_{i, j >= 1} 1/(i^3*j^2*binomial(i+j, i)). More generally, for n >= 2, Sum_{i, j >= 1} 1/(i^n*j^2*binomial(i+j, i)) = zeta(2)*zeta(n) - zeta(n+2) - zeta(n,2). - Peter Bala, Aug 05 2025

A258988 Decimal expansion of the multiple zeta value (Euler sum) zetamult(4,3).

Original entry on oeis.org

0, 8, 5, 1, 5, 9, 8, 2, 2, 5, 3, 4, 8, 3, 3, 6, 5, 1, 4, 0, 6, 8, 0, 6, 0, 1, 8, 8, 7, 2, 3, 6, 7, 3, 4, 5, 9, 5, 7, 3, 3, 9, 5, 0, 8, 5, 8, 6, 8, 7, 7, 3, 2, 0, 4, 6, 7, 1, 0, 3, 4, 3, 2, 0, 5, 3, 3, 0, 8, 5, 7, 6, 7, 5, 0, 8, 7, 1, 7, 6, 6, 5, 1, 1, 1, 7, 3, 3, 8, 6, 7, 5, 8, 1, 8, 5, 0, 2, 0, 7, 2, 0, 5, 4, 1

Offset: 0

Jean-François Alcover, Jun 16 2015

			0.0851598225348336514068060188723673459573395085868773204671034320533...

Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function

Cf. A072691 (zetamult(1,1)), A197110 (zetamult(2,2)), A258983 (zetamult(3,2)), A258984 (4,2), A258985 (5,2), A258947 (6,2), A258986 (2,3), A258987 (3,3), A258982 (5,3), A258989 (2,4), A258990 (3,4), A258991 (4,4).

Join[{0}, RealDigits[17*Zeta[7] - 10*Zeta[2]*Zeta[5], 10, 104] // First]

zetamult([4,3]) \\ Charles R Greathouse IV, Jan 21 2016

zetamult(4,3) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^2*n^3)) = 17*zeta(7) - 10*zeta(2)*zeta(5).

A258990 Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,4).

Original entry on oeis.org

2, 0, 7, 5, 0, 5, 0, 1, 4, 6, 1, 5, 7, 3, 2, 0, 9, 5, 9, 0, 7, 8, 0, 7, 6, 0, 5, 4, 9, 4, 6, 7, 1, 4, 6, 5, 4, 4, 1, 8, 2, 8, 6, 7, 9, 5, 5, 0, 6, 0, 6, 1, 9, 0, 4, 1, 9, 5, 1, 7, 8, 9, 6, 5, 6, 9, 7, 1, 0, 1, 1, 9, 9, 7, 1, 6, 0, 7, 8, 0, 0, 7, 8, 0, 9, 8, 6, 6, 4, 3, 6, 3, 3, 0, 5, 2, 3, 0, 2, 0, 2, 9, 6, 5, 9

Offset: 0

Jean-François Alcover, Jun 16 2015

			0.20750501461573209590780760549467146544182867955060619041951789656971...

Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function

Cf. A072691 (zetamult(1,1)), A197110 (zetamult(2,2)), A258983 (zetamult(3,2)), A258984 (4,2), A258985 (5,2), A258947 (6,2), A258986 (2,3), A258987 (3,3), A258988 (4,3), A258982 (5,3), A258989 (2,4), A258991 (4,4).

RealDigits[10*Zeta[2]*Zeta[5] + Zeta[3]*Zeta[4] - 18*Zeta[7], 10, 105] // First

zetamult([3,4]) \\ Charles R Greathouse IV, Jan 21 2016

zetamult(3,4) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^3*n^4)) = 10*zeta(2)*zeta(5) + zeta(3)*zeta(4) - 18*zeta(7).

A258991 Decimal expansion of the multiple zeta value (Euler sum) zetamult(4,4).

Original entry on oeis.org

0, 8, 3, 6, 7, 3, 1, 1, 3, 0, 1, 6, 4, 9, 5, 3, 6, 1, 6, 1, 4, 8, 9, 0, 4, 3, 6, 5, 4, 2, 3, 8, 7, 7, 0, 5, 4, 3, 8, 2, 4, 6, 7, 3, 2, 5, 5, 4, 1, 5, 4, 1, 6, 8, 3, 6, 0, 7, 5, 9, 1, 8, 3, 5, 5, 4, 3, 8, 1, 9, 1, 2, 7, 1, 4, 5, 6, 2, 4, 0, 1, 1, 9, 9, 6, 0, 7, 2, 6, 9, 1, 9, 7, 6, 9, 7, 6, 6, 4, 2, 6, 0, 3, 7, 6, 9, 7

Offset: 0

Jean-François Alcover, Jun 16 2015

			0.08367311301649536161489043654238770543824673255415416836075918355438...

Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function

Cf. A072691 (zetamult(1,1)), A197110 (zetamult(2,2)), A258983 (zetamult(3,2)), A258984 (4,2), A258985 (5,2), A258947 (6,2), A258986 (2,3), A258987 (3,3), A258988 (4,3), A258982 (5,3), A258989 (2,4), A258990 (3,4).

Join[{0}, RealDigits[(1/2)*(Zeta[4]^2 - Zeta[8]), 10, 106] // First]

zetamult([4,4]) \\ Charles R Greathouse IV, Jan 21 2016

(zeta(4)^2-zeta(8))/2 \\ Charles R Greathouse IV, Jan 20 2022

zetamult(4,4) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^4*n^4)) = (1/2)*(zeta(4)^2 - zeta(8)).

A025281 a(n) = sopfr(n!), where sopfr = A001414 is the integer log.

Original entry on oeis.org

0, 0, 2, 5, 9, 14, 19, 26, 32, 38, 45, 56, 63, 76, 85, 93, 101, 118, 126, 145, 154, 164, 177, 200, 209, 219, 234, 243, 254, 283, 293, 324, 334, 348, 367, 379, 389, 426, 447, 463, 474, 515, 527, 570, 585, 596, 621, 668, 679, 693, 705, 725, 742, 795, 806, 822, 835, 857, 888

Offset: 0

Wouter Meeussen

nonn

József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter IV, p. 144.

Chai Wah Wu, Table of n, a(n) for n = 0..10000 (terms n = 0..1000 from T. D. Noe)
Krishnaswami Alladi and Paul Erdős, On an additive arithmetic function, Pacific Journal of Mathematics, Vol. 71, No. 2 (1977), pp. 275-294; alternative link.

Partial sums of A001414.

Cf. A000142, A072691.

a:= proc(n) option remember; `if`(n<2, 0,
      a(n-1)+add(i[1]*i[2], i=ifactors(n)[2]))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Apr 09 2021

sopfr[n_] := Plus @@ Times @@@ FactorInteger@ n; a[n_] := a[n] = a[n - 1] + sopfr[n]; a[0] = a[1] = 0; Array[a, 59, 0] (* Robert G. Wilson v, May 18 2015 *)

for(n=1,100,print1(sum(k=1,n,sum(i=1,omega(k), component(component(factor(k),1),i)*component(component(factor(k),2),i))),","))

from sympy import factorial, factorint
def A025281(n): return sum(p*e for p, e in factorint(factorial(n)).items()) # Chai Wah Wu, Apr 09 2021

a(n) = A001414(A000142(n)).
From Benoit Cloitre, Apr 14 2002: (Start)
a(0)=0; for n>0, a(n) = Sum_{k=1..n} A001414(k).
Asymptotic formula: a(n) ~ (Pi^2/12)*n^2/log(n). [Proven by Alladi and Erdős (1977). - Amiram Eldar, Mar 04 2021]
(End)

A107758 (+2)Sigma(n): If n = Product p_i^r_i then a(n) = Product (2 + Sum p_i^s_i, s_i=1 to r_i) = Product (1 + (p_i^(r_i+1)-1)/(p_i-1)), a(1) = 1.

Original entry on oeis.org

1, 4, 5, 8, 7, 20, 9, 16, 14, 28, 13, 40, 15, 36, 35, 32, 19, 56, 21, 56, 45, 52, 25, 80, 32, 60, 41, 72, 31, 140, 33, 64, 65, 76, 63, 112, 39, 84, 75, 112, 43, 180, 45, 104, 98, 100, 49, 160, 58, 128, 95, 120, 55, 164, 91, 144, 105, 124, 61, 280, 63, 132, 126, 128, 105, 260, 69, 152, 125, 252, 73, 224, 75, 156, 160

Offset: 1

Yasutoshi Kohmoto, May 25 2005

			a(6) = (2+2)*(2+3) = 20.

Daniel Suteu, Table of n, a(n) for n = 1..10000
Sagar Mandal, Divisibility and Sequence Properties of sigma+ and phi+, arXiv:2508.11660 [math.GM], 2025.
József Sándor and Krassimir Atanassov, Some new arithmetic functions, Notes on Number Theory and Discrete Mathematics, Volume 30, 2024, Number 4, Pages 851-856. See sigma+ function.

Cf. A000203, A055653, A072691, A107759.

Cf. A052396 (k such that a(k) = 2k), A387720 (k such that a(k) < 2k), A387721 (k such that a(k) > 2k), A387725, A386390 (k such that k-1 | a(k)).

A107758 := proc(n) local pf,p ; if n = 1 then 1; else pf := ifactors(n)[2] ; mul( 1+(op(1,p)^(op(2,p)+1)-1)/(op(1,p)-1), p=pf) ; end if; end proc:
seq(A107758(n),n=1..60) ; # R. J. Mathar, Jan 07 2011

Table[DivisorSum[n, DivisorSigma[1, #] &, CoprimeQ[n/#, #] &], {n, 54}] (* Michael De Vlieger, Jun 27 2018 *)
f[p_, e_] := 1 + (p^(e + 1) - 1)/(p - 1); a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Aug 26 2022 *)

a(n) = sumdiv(n, d, if(gcd(n/d, d) == 1, sigma(d))); \\ Daniel Suteu, Jun 27 2018

A107758(n) =  { my(f = factor(n)); prod(k=1, #f~, 1+sigma(f[k, 1]^f[k, 2])); }; \\ Antti Karttunen, Sep 06 2025

a(n) = Sum_{d|n, gcd(n/d, d) = 1} sigma(d), where sigma(d) is the sum of the divisors of d. - Daniel Suteu, Jun 27 2018

Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 + 1/p^2 - 1/p^3) = 1.0741158... . - Amiram Eldar, Nov 01 2022

More terms from Antti Karttunen, Sep 06 2025

A244583 a(n) = sum of all divisors of all positive integers <= prime(n).

Original entry on oeis.org

4, 8, 21, 41, 99, 141, 238, 297, 431, 690, 794, 1136, 1384, 1524, 1806, 2304, 2846, 3076, 3699, 4137, 4406, 5128, 5645, 6499, 7755, 8401, 8721, 9393, 9783, 10513, 13280, 14095, 15443, 15871, 18232, 18756, 20320, 21873, 22875, 24604, 26274, 27002, 29982, 30684

Offset: 1

Omar E. Pol, Jun 30 2014

nonn

Limit_{n->oo} a(n)/prime(n)^2 = zeta(2)/2 = Pi^2/12 = A072691 = 0.82246703342.... For example, at n = 2*10^6, the ratio converges to 0.822467033... (+-2 in the last digit with increments on n of +100). If the ratio is calculated with a nonprime for the upper summation limit then the ratio runs slightly larger and converges slower. See formula section of A024916 for the general case. - Richard R. Forberg, Jan 04 2015

This is a subsequence of A024916 therefore a(n) also has a symmetric representation. For more information see A236104, A237593. - Omar E. Pol, Jan 05 2015

Cf. A000040, A000203, A024916, A236104, A237593, A237270, A237271, A244576, A244578.

Partial sums of A358683.

a244583[n_] := Sum[DivisorSigma[1, i], {i, #}] & /@ Prime[Range@n]; a244583[44] (* Michael De Vlieger, Jan 06 2015 *)

a(n) = sum(i=1, prime(n), sigma(i)); \\ Michel Marcus, Sep 29 2014

from math import isqrt
from sympy import prime
def A244583(n): return -(s:=isqrt(p:=prime(n)))**2*(s+1) + sum((q:=p//k)*((k<<1)+q+1) for k in range(1,s+1))>>1 # Chai Wah Wu, Oct 23 2023

a(n) = A024916(A000040(n)).

a(n) = A001248(n) - A050482(n). - Omar E. Pol, Jan 05 2015

More terms from Michel Marcus, Sep 29 2014

A275703 Decimal expansion of the Dirichlet eta function at 6.

Original entry on oeis.org

9, 8, 5, 5, 5, 1, 0, 9, 1, 2, 9, 7, 4, 3, 5, 1, 0, 4, 0, 9, 8, 4, 3, 9, 2, 4, 4, 4, 8, 4, 9, 5, 4, 2, 6, 1, 4, 0, 4, 8, 8, 5, 6, 9, 3, 4, 6, 9, 3, 2, 6, 8, 8, 8, 0, 3, 4, 8, 3, 3, 3, 9, 3, 2, 5, 4, 1, 9, 6, 8, 0, 2, 1, 8, 6, 2, 7, 1, 7, 1, 3, 5, 7, 3, 9, 3, 7, 2, 9, 1, 1, 2, 7, 9, 5, 5, 9, 4, 6, 4

Offset: 0

Terry D. Grant, Aug 05 2016

It appears that each sum of a Dirichlet eta function is 1/2^(x-1) less than the zeta(x), where x is a positive integer > 1. In this case, eta(x) = eta(6) = (31/32)*zeta(6) = 31*(Pi^6)/30240. Therefore eta(6) = 1/2^(6-1) or 1/32nd less than zeta(6) (see A013664). [Edited by Petros Hadjicostas, May 07 2020]

			31*(Pi^6)/30240 = 0.9855510912974...

Cf. A002162 (decimal expansion of value at 1), A072691 (value at 2), A197070 (value at 3), A267315 (value at 4), A267316 (value at 5), A275710 (value at 7).

Cf. A013664, A136677, A334605.

Mathematica
```
RealDigits[31*(Pi^6)/30240,10,100]
```

s = RLF(0); s
RealField(110)(s)
for i in range(1, 10000): s -= (-1)^i / i^6
print(s) # Terry D. Grant, Aug 05 2016

eta(6) = 31*(Pi^6)/30240 = 31*A092732/30240 = Sum_{n>=1} (-1)^(n+1)/n^6.

eta(6) = lim_{n -> infinity} A136677(n)/A334605(n). - Petros Hadjicostas, May 07 2020

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A258984 Decimal expansion of the multiple zeta value (Euler sum) zetamult(4,2).

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A258985 Decimal expansion of the multiple zeta value (Euler sum) zetamult(5,2).

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A258986 Decimal expansion of the multiple zeta value (Euler sum) zetamult(2,3).

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A258988 Decimal expansion of the multiple zeta value (Euler sum) zetamult(4,3).

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A258990 Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,4).

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A258991 Decimal expansion of the multiple zeta value (Euler sum) zetamult(4,4).

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A025281 a(n) = sopfr(n!), where sopfr = A001414 is the integer log.

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