A072691 -id:A072691 - cp's OEIS frontend

A258987 Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,3).

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

Offset: 0

Jean-François Alcover, Jun 16 2015

			0.213798868224592547099583574508033649640958957865517556144512748947...

Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function
Index to constants which are multiple zeta values (3,3)

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), A258988 (4,3), A258982 (5,3), A258989 (2,4), A258990 (3,4), A258991 (4,4).

RealDigits[(1/2)*Zeta[3]^2 - (1/2)*Zeta[6], 10, 103] // First (* Corrected by Detlef Meya, Jun 06 2025 *)

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

zetamult(3,3) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^3*n^3)) = (1/2)*zeta(3)^2 - (1/2)*zeta(6). - [Corrected by Detlef Meya, Jun 06 2025 ]

A353908 Decimal expansion of Pi^2/36.

Original entry on oeis.org

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

Offset: 0

Omar E. Pol, May 10 2022

Ratio between the volume of the stepped pyramid with an infinite number of levels described in A245092 and that of the circumscribed cube (see the first formula).
See also Vaclav Kotesovec's formula (2016) in A175254.
Volume shared by a sphere inscribed in a cube of volume Pi and one of the six pyramids inscribed in the cube. - Omar E. Pol, Sep 01 2024

			0.2741556778080377394120691944410041982031583168677997396225930382283345784...

Index entries for transcendental numbers

Cf. A000203, A000796, A002388, A013661, A019673, A021040, A024916, A072691, A086463, A086729, A091476, A100044, A102753, A175254, A195055, A237271, A237593, A245092.

evalf(Pi^2/36, 121);  # Alois P. Heinz, May 11 2022

RealDigits[Pi^2/36, 10, 100][[1]] (* Amiram Eldar, May 11 2022 *)

PARI
```
Pi^2/36
```
PARI
```
zeta(2)/6
```

Equals lim_{n->oo} A175254(n)/n^3.
Equals A002388/36.
Equals A102753/18.
Equals A195055/12.
Equals A091476/9.
Equals A013661/6.
Equals A100044/4.
Equals A072691/3.
Equals A086463/2.
Equals A086729*2.
Equals A019673^2.
Equals A002388*A021040.
Equals Re(dilog((1+sqrt(3)*i)/2)). - Mohammed Yaseen, Jul 03 2024

A224880 a(n) = 2n + sum of divisors of n.

Original entry on oeis.org

3, 7, 10, 15, 16, 24, 22, 31, 31, 38, 34, 52, 40, 52, 54, 63, 52, 75, 58, 82, 74, 80, 70, 108, 81, 94, 94, 112, 88, 132, 94, 127, 114, 122, 118, 163, 112, 136, 134, 170, 124, 180, 130, 172, 168, 164, 142, 220, 155, 193, 174, 202, 160, 228, 182, 232, 194, 206

Offset: 1

Hans Havermann, Jul 23 2013

nonn

This sequence is A033880 for the negative integers, thus making explicit the mapping noted in A075701.
From Omar E. Pol, Jun 21 2018: (Start)
a(n) is also the total area of the terraces and the vertical sides that are visible in the perspective view at the n-th level (starting from the top) of the stepped pyramid described in A245092.
Partial sums give A299692. (End)

			a(6) = 2*6 + (1+2+3+6) = 24.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000203, A033879, A033880, A072691, A075701, A155085, A237593, A245092, A299692.

with(numtheory); seq(2*k+sigma(k),k=1..100); # Wesley Ivan Hurt, Jul 24 2013

Mathematica
```
Table[2*n+DivisorSigma[1,n],{n,64}]
```

vector(80, n, 2*n + sigma(n)) \\ Michel Marcus, Aug 19 2015

a(n) = A155085(n) + n.
a(n) = 2n + sigma(n) = A005843(n) + A000203(n) = A033879(n) + 2*A000203(n) = A033880(n) + 2*A005843(n) = 2*A155085(n) - A000203(n) = 2*A000203(n) - A033880(n). - Wesley Ivan Hurt, Jul 24 2013
G.f.: 2*x/(1 - x)^2 + Sum_{k>=1} x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Mar 17 2017
a(n) = A001065(n) + A008585(n). - Omar E. Pol, Mar 06 2018
Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = zeta(2)/2 + 1 = A072691 + 1 = 1.822467... . - Amiram Eldar, Mar 17 2024

A258982 Decimal expansion of the multiple zeta value (Euler sum) zetamult(5,3).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 16 2015

			0.03770767298484754401130478229365991482260131941527752401264507780391...

Richard E. Crandall, Joe P. Buhler, On the evaluation of Euler Sums, Exp. Math. 3 (4) (1994) 275-285 Table 1.
Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function

Cf. A072691, A197110, A258947.

digits = 99; zetamult[6, 2] = NSum[HarmonicNumber[m-1, 2]/m^6, {m, 2, Infinity}, WorkingPrecision -> digits+20, NSumTerms -> 200, Method -> {"NIntegrate", "MaxRecursion" -> 18}]; zetamult[5, 3] = 5*Zeta[3]*Zeta[5] - (147/24)*Zeta[8] - (5/2)*zetamult[6, 2]; Join[{0}, RealDigits[zetamult[5, 3], 10, digits] // First]

zetamult([5,3]) \\ Charles R Greathouse IV, Jan 21 2016

zetamult(5,3) = Sum_{m>=2} (sum_{n=1..m-1} 1/(m^5*n^3)).
Equals Sum_{m>=2} (H(m-1, 3)+polygamma(2,1)/2+zeta(3))/m^5, where H(n,3) is the n-th harmonic number of order 3.
Also equals Sum_{m>=2} (polygamma(2,m)+zeta(3))/(2m^5).
Also equals 5*zeta(3)*zeta(5) - (147/24)*zeta(8) - (5/2)*zetamult(6, 2), where zetamult(6,2) is A258947.

A258983 Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,2).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 16 2015

Also zetamult(2, 2, 1). - Charles R Greathouse IV, Jan 04 2017

			0.2288103976033537597687461489416887919325093427198821602294071...

Dominique Manchon, Arborified multiple zeta values, arXiv:1603.01498 [math.CO], 2016.
Jonathan Borwein and Roland Girgensohn, Evaluation of triple Euler Sums, Elec. Jour. of Comb., Vol. 3, Issue 1, 1996. Article R23 (see page 21).
Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function

Cf. A072691 (zetamult(1,1)), A197110 (zetamult(2,2)), 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), A258991 (4,4).

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

RealDigits[3*Zeta[2]*Zeta[3] - (11/2)*Zeta[5], 10, 104] // First

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

zetamult([2,2,1]) \\ Charles R Greathouse IV, Jan 04 2017

Equals Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^3*n^2)) = 3*zeta(2)*zeta(3) - (11/2)*zeta(5).

A258989 Decimal expansion of the multiple zeta value (Euler sum) zetamult(2,4).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 16 2015

			0.67452391403396814049156060825742993927838436513788957970691722144377...

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), A258990 (3,4), A258991 (4,4).

RealDigits[(25/12)*Zeta[6] - Zeta[3]^2, 10, 103] // First

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

zetamult(2,4) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^2*n^4)) = (25/12)*zeta(6) - zeta(3)^2.

Equals Sum_{i, j >= 1} 1/(i^4*j^2*binomial(i+j, i)). - Peter Bala, Aug 05 2025

A267316 Decimal expansion of the Dirichlet eta function at 5.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jan 13 2016

			1/1^5 - 1/2^5 + 1/3^5 - 1/4^5 + 1/5^5 - 1/6^5 + ... = 0.972119770446909305935655143553469532553513362...

OEIS Wiki, Euler's alternating zeta function
Eric Weisstein's World of Mathematics, Dirichlet Eta Function
Wikipedia, Dirichlet Eta Function

Cf. A002162 (value at 1), A013663, A072691 (value at 2), A197070 (value at 3), A267315 (value at 4), A136676, A334604.

RealDigits[(15 Zeta[5])/16, 10, 120][[1]]

15*zeta(5)/16 \\ Michel Marcus, Feb 01 2016

s = RLF(0); s
RealField(110)(s)
for i in range(1, 10000): s += -((-1)^i/((i)^5))
print(s) # Terry D. Grant, Aug 05 2016

Equals Sum_{k > 0} (-1)^(k+1)/k^5 = (15*zeta(5))/16.

Equals Lim_{n -> infinity} A136676(n)/A334604(n). - Petros Hadjicostas, May 07 2020

A072692 Sum of sigma(j) for 1<=j<=10^n, where sigma(j) is the sum of the divisors of j.

Original entry on oeis.org

1, 87, 8299, 823081, 82256014, 8224740835, 822468118437, 82246711794796, 8224670422194237, 822467034112360628, 82246703352400266400, 8224670334323560419029, 822467033425357340138978, 82246703342420509396897774, 8224670334241228180927002517

Offset: 0

Rick L. Shepherd, Jul 02 2002

nonn

			For n=1, the sum of sigma(j) for j<=10 is 1+3+4+7+6+12+8+15+13+18=87, so a(1)=87 (=69+18=A049000(1)+A046915(1)).

P. L. Patodia, Seth Troisi and Hiroaki Yamanouchi, Table of n, a(n) for n = 0..36 (terms a(0)-a(18) by P. L. Patodia and a(19)-a(24) by Seth Troisi)
Leonhard Euler, Découverte d'une loi tout extraordinaire des nombres par rapport à la somme de leurs diviseurs, 1747, The Euler Archive, (Eneström Index) E175.
P. L. Patodia (pannalal(AT)usa.net), PARI program for A072692 and A024916

Compare with A049000. Note that a(n) = A049000(n) + A046915(n).

Cf. A000203 (sigma(n)), A072691 (Pi^2/12), A049000, A046915, A024916, A025281.

for(m=0,10,print1(sum(n=1,k=10^m,n*(k\n)),",")) \\ Improved by M. F. Hasler, Apr 18 2015

A072692(n)=A024916(10^n) \\ This is very efficient, using efficient code of A024916. - M. F. Hasler, Apr 18 2015

[(i, sum([d*(10**i//d) for d in range(1,10**i+1)])) for i in range(8)] # Seth A. Troisi, Jun 27 2010

from math import isqrt
def A072692(n): return -(s:=isqrt(m:=10**n))**2*(s+1)+sum((q:=m//k)*((k<<1)+q+1) for k in range(1,s+1))>>1 # Chai Wah Wu, Oct 23 2023

Asymptotic formula: a(n) ~ Pi^2/12 * 10^2n. See A072691 for Pi^2/12. Observe that A025281 also contains that constant in its asymptotic formula.

More terms from P L Patodia (pannalal(AT)usa.net), Jan 11 2008, Jun 25 2008

Corrected by N. J. A. Sloane, Jun 08 2008, following suggestions from Don Reble and David W. Wilson

A107759 a(n) = (+2)UnitarySigma(n): if n = Product p_i^r_i then a(n) = Product (2 + p_i^r_i).

Original entry on oeis.org

1, 4, 5, 6, 7, 20, 9, 10, 11, 28, 13, 30, 15, 36, 35, 18, 19, 44, 21, 42, 45, 52, 25, 50, 27, 60, 29, 54, 31, 140, 33, 34, 65, 76, 63, 66, 39, 84, 75, 70, 43, 180, 45, 78, 77, 100, 49, 90, 51, 108, 95, 90, 55, 116, 91

Offset: 1

Yasutoshi Kohmoto, May 25 2005

			a(12) = (2+3)*(2+4) = 30.

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

Cf. A007429, A034448, A054862, A072691, A107758, A330594.

A107759 := proc(n) local pf,p ; if n = 1 then 1; else pf := ifactors(n)[2] ; mul( 2+op(1,p)^op(2,p), p=pf) ; end if; end proc:
seq(A107759(n),n=1..60) ; # R. J. Mathar, Jan 07 2011

a[1] = 1; a[n_] := Times @@ (2 + Power @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 20 2020 *)

a(n) = Sum_{d|n, gcd(d, n/d) = 1} usigma(d), where usigma = A034448. - Ilya Gutkovskiy, Mar 27 2020

Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 + 1/p^2 - 2/p^3) = A072691 * A330594 = 0.910438... . - Amiram Eldar, Nov 01 2022

A162395 a(n) = -(-1)^n * n^2.

Original entry on oeis.org

1, -4, 9, -16, 25, -36, 49, -64, 81, -100, 121, -144, 169, -196, 225, -256, 289, -324, 361, -400, 441, -484, 529, -576, 625, -676, 729, -784, 841, -900, 961, -1024, 1089, -1156, 1225, -1296, 1369, -1444, 1521, -1600, 1681, -1764, 1849, -1936, 2025, -2116, 2209, -2304, 2401, -2500

Offset: 1

Michael Somos, Jul 02 2009

This sequence is the denominator of (Pi^2)/12 = 1/1-1/4+1/9-1/16+1/25-1/36+... - Mohammad K. Azarian, Dec 29 2011

Also, circulant determinant of [1,2,...,n,n-1,...,1], i.e., determinant of the (2n-1) X (2n-1) matrix which has this as first row (and also first column), where row k+1 is obtained by cyclically shifting row k one place to the left. - M. F. Hasler, Dec 17 2016

			G.f. = x - 4*x^2 + 9*x^3 - 16*x^4 + 25*x^5 - 36*x^6 + 49*x^7 - 64*x^8 + 81*x^9 + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Michael Somos, Rational Function Multiplicative Coefficients.
Index entries for linear recurrences with constant coefficients, signature (-3,-3,-1).

Cf. A000290, A072691.

For the reversion of this sequence see A263843 (and also A007297).

[(-1)^(n+1) * n^2: n in [1..60]]; // Vincenzo Librandi, Feb 15 2013

Table[(-1)^(n+1) * n^2, {n, 60}] (* Vincenzo Librandi, Feb 15 2013 *)

PARI
```
{a(n) = -(-1)^n * n^2};
```

Euler transform of length 2 sequence [-4, 3].
a(n) is multiplicative with a(2^e) = -(4^e) if e>0, a(p^e) = (p^2)^e if p>2.
G.f.: x * (1 - x) / (1 + x)^3.
E.g.f.: exp(-x) * (x - x^2).
a(n) = a(-n) = -(-1)^n * A000290(n) for all n in Z.
Sum_{n>=1} 1/a(n) = Pi^2/12 (A072691). - Amiram Eldar, Dec 10 2022
Dirichlet g.f.: zeta(s-2)*(1-2^(3-s)) = DirichletEta(s-2). - Amiram Eldar, Jan 07 2023

cp's OEIS Frontend

A258987 Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,3).

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A353908 Decimal expansion of Pi^2/36.

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A224880 a(n) = 2n + sum of divisors of n.

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A258982 Decimal expansion of the multiple zeta value (Euler sum) zetamult(5,3).

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A258983 Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,2).

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A258989 Decimal expansion of the multiple zeta value (Euler sum) zetamult(2,4).

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A267316 Decimal expansion of the Dirichlet eta function at 5.

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