A010057 -id:A010057 - cp's OEIS frontend

A013675 Decimal expansion of zeta(17).

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

Offset: 1

N. J. A. Sloane

			1.0000076371976378997622736002935630292130882490902626790953798439729356...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 811.

Cf. A013663, A013667, A013669, A013671, A013675, A013677, A010057.

RealDigits[Zeta[17], 10, 75][[1]] (* Vincenzo Librandi, Feb 24 2015 *)

zeta(17) \\ Charles R Greathouse IV, Dec 04 2013

From Peter Bala, Dec 04 2013: (Start)
Definition: zeta(17) = sum {n >= 1} 1/n^17.
zeta(17) = 2^17/(2^17 - 1)*( sum {n even} n^11*p(n)*p(1/n)/(n^2 - 1)^18 ), where p(n) = n^8 + 36*n^6 + 126*n^4 + 84*n^2 + 9. Cf. A013663, A013667 and A013671.
(End)
zeta(17) = Sum_{n >= 1} (A010052(n)/n^(17/2)) = Sum_{n >= 1} ( (floor(sqrt(n)) - floor(sqrt(n-1)))/n^(17/2) ). - Mikael Aaltonen, Feb 23 2015
zeta(17) = Product_{k>=1} 1/(1 - 1/prime(k)^17). - Vaclav Kotesovec, May 02 2020

A051343 Number of ways of writing n as a sum of 3 nonnegative cubes (counted naively).

Original entry on oeis.org

1, 3, 3, 1, 0, 0, 0, 0, 3, 6, 3, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 1, 0, 0, 3, 6, 3, 0, 0, 0, 0, 0, 6, 6, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 3, 0, 3, 6, 3, 0, 0, 0, 0, 0, 6, 6, 0, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 6

Offset: 0

N. J. A. Sloane

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A051344.

Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.

Maple
```
series(add(x^(n^3), n=0..10)^3,x,1000);
```

first(n)=my(s=vector(n+1)); for(k=0,sqrtnint(n,3), s[k^3+1]=1); Vec(Ser(s,,n+1)^3) \\ Charles R Greathouse IV, Sep 16 2016

A113061 Sum of cube divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 28, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 28, 1, 9, 1, 1, 1, 1, 1, 1, 1, 73, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 28, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9

Offset: 1

Paul Barry, Oct 13 2005

Multiplicative with a(p^e) = (p^(3*(1+floor(e/3)))-1)/(p^3-1). The Dirichlet generating function is zeta(s)*zeta(3s-3). The sequence is the inverse Mobius transform of n*A010057(n). - R. J. Mathar, Feb 18 2011

Antti Karttunen, Table of n, a(n) for n = 1..19683
R. J. Mathar, Survey of Dirichlet series of multiplicative arithmetic functions, arXiv:1106.4038 [math.NT] (2011), Remark 15.
Index entries for sequences related to sums of divisors

Cf. A004709, A010057, A035316.

A113061 := proc(n)
    local a,pe,p,e;
    a := 1;
    for pe in ifactors(n)[2] do
        p := pe[1] ;
        e := pe[2] ;
        e := 3*(1+floor(e/3)) ;
        a := a*(p^e-1)/(p^3-1) ;
    end do:
    a ;
end proc:
seq(A113061(n),n=1..100) ; # R. J. Mathar, Oct 08 2017

a[n_] := Sum[If[IntegerQ[d^(1/3)], d, 0], {d, Divisors[n]}];
Array[a, 100] (* Jean-François Alcover, Nov 25 2017 *)
f[p_, e_] := (p^(3*(1 + Floor[e/3])) - 1)/(p^3 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 01 2020 *)

A113061(n) = sumdiv(n,d,ispower(d,3)*d); \\ Antti Karttunen, Oct 08 2017

;; With memoization-macro definec, after the multiplicative formula of R. J. Mathar:
(definec (A113061 n) (if (= 1 n) n (let ((p (A020639 n)) (e (A067029 n))) (* (/ (+ -1 (expt p (* 3 (+ 1 (A002264 e))))) (+ -1 (expt p 3))) (A113061 (A028234 n)))))) ;; Antti Karttunen, Oct 08 2017

G.f.: Sum_{k>=1} k^3*x^(k^3)/(1 - x^(k^3)). - Ilya Gutkovskiy, Dec 24 2016
a(n) = Sum_{d|n} A010057(d)*d. - Antti Karttunen, Oct 08 2017
Sum_{k=1..n} a(k) ~ zeta(4/3)*n^(4/3)/4 - n/2. - Vaclav Kotesovec, Dec 01 2020

A173677 Number of ways of writing n as a sum of two nonnegative cubes.

Original entry on oeis.org

1, 2, 1, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 0

N. J. A. Sloane, Nov 24 2010

Order matters. This is the coefficient of q^n in the expansion of {Sum_{m>=0} q^(m^3)}^2.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A004829, A008451, A010057, A051343, A173676.

Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.

list(n)=my(q='q);Vec(sum(m=0,(n+.5)^(1/3),q^(m^3),O(q^(n+1)))^2) \\ Charles R Greathouse IV, Jun 07 2012

a(n) = Sum_{k=0..n} c(k) * c(n-k), where c = A010057. - Wesley Ivan Hurt, Nov 09 2023

A173682 Number of ways of writing n as a sum of 9 nonnegative cubes.

Original entry on oeis.org

1, 9, 36, 84, 126, 126, 84, 36, 18, 73, 252, 504, 630, 504, 252, 72, 45, 252, 756, 1260, 1260, 756, 252, 36, 84, 504, 1260, 1689, 1332, 756, 588, 630, 630, 882, 1332, 1341, 1134, 1638, 2520, 2520, 1638, 1008, 828, 756, 1638, 3780, 5040, 3780, 1596, 504, 252, 588, 2520, 5040, 5076, 2772, 1296, 1332, 1296, 1386, 2772, 3816, 2772, 2142, 3798, 5121, 4032

Offset: 0

N. J. A. Sloane, Nov 25 2010

nonn

Order matters. This is the coefficient of q^n in the expansion of {Sum_{m>=0} q^(m^3)}^9.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A173677.

Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.

A173676 Number of ways of writing n as a sum of seven nonnegative cubes.

Original entry on oeis.org

1, 7, 21, 35, 35, 21, 7, 1, 7, 42, 105, 140, 105, 42, 7, 0, 21, 105, 210, 210, 105, 21, 0, 0, 35, 140, 210, 147, 77, 105, 140, 105, 77, 112, 105, 77, 210, 420, 420, 210, 63, 42, 21, 105, 420, 630, 420, 105, 7, 7, 0, 140, 420, 420, 161, 105, 211, 210, 105, 126, 210, 105, 105, 420, 637, 462, 210, 182, 147, 42, 217, 630, 672, 420, 420, 427, 210, 42

Offset: 0

N. J. A. Sloane, Nov 24 2010

Order matters. This is the coefficient of q^n in the expansion of {Sum_{m>=0} q^(m^3)}^7.

It is known that a(n)>0 if n is even and > 454.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
N. D. Elkies, Every even number greater than 454 is the sum of seven cubes, arXiv:1009.3983

Cf. A004829, A008451.

Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.

A173678 Number of ways of writing n as a sum of 4 nonnegative cubes.

Original entry on oeis.org

1, 4, 6, 4, 1, 0, 0, 0, 4, 12, 12, 4, 0, 0, 0, 0, 6, 12, 6, 0, 0, 0, 0, 0, 4, 4, 0, 4, 12, 12, 4, 0, 1, 0, 0, 12, 24, 12, 0, 0, 0, 0, 0, 12, 12, 0, 0, 0, 0, 0, 0, 4, 0, 0, 6, 12, 6, 0, 0, 0, 0, 0, 12, 12, 4, 12, 12, 4, 0, 0, 6, 0, 12, 24, 12, 0, 0, 0, 0, 0, 12, 16, 4, 0, 0, 0, 0, 0, 4, 4, 0, 12, 24, 12, 0, 0, 0, 0, 0, 24, 24, 0, 0, 0, 0, 0, 0, 12, 1, 0, 0

Offset: 0

N. J. A. Sloane, Nov 24 2010

Order matters. This is the coefficient of q^n in the expansion of {Sum_{m>=0} q^(m^3)}^4.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A004829, A008451, A010057, A173677, A051343, A173676.
Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.
Without order you get A025448.

A173679 Number of ways of writing n as a sum of 5 nonnegative cubes.

Original entry on oeis.org

1, 5, 10, 10, 5, 1, 0, 0, 5, 20, 30, 20, 5, 0, 0, 0, 10, 30, 30, 10, 0, 0, 0, 0, 10, 20, 10, 5, 20, 30, 20, 5, 5, 5, 0, 20, 60, 60, 20, 0, 1, 0, 0, 30, 60, 30, 0, 0, 0, 0, 0, 20, 20, 0, 10, 30, 30, 10, 0, 5, 0, 0, 30, 60, 35, 20, 30, 20, 5, 0, 30, 30, 20, 60, 60, 20, 0, 0, 10, 0, 30, 70, 50, 10, 0, 0, 0, 0, 20, 40, 20, 20, 60, 60, 20, 0, 5, 10, 0, 60, 120

Offset: 0

N. J. A. Sloane, Nov 25 2010

nonn

Order matters. This is the coefficient of q^n in the expansion of {Sum_{m>=0} q^(m^3)}^5.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A173677.

Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.

A173680 Number of ways of writing n as a sum of 6 nonnegative cubes.

Original entry on oeis.org

1, 6, 15, 20, 15, 6, 1, 0, 6, 30, 60, 60, 30, 6, 0, 0, 15, 60, 90, 60, 15, 0, 0, 0, 20, 60, 60, 26, 30, 60, 60, 30, 21, 30, 15, 30, 120, 180, 120, 30, 6, 6, 0, 60, 180, 180, 60, 0, 1, 0, 0, 60, 120, 60, 15, 60, 90, 60, 15, 30, 30, 0, 60, 180, 186, 90, 60, 66, 30, 6, 90, 180, 120, 120, 180, 120, 30, 0, 60, 60, 60, 200, 240, 120, 20, 0, 15, 0, 60, 180, 180

Offset: 0

N. J. A. Sloane, Nov 25 2010

nonn

Order matters. This is the coefficient of q^n in the expansion of {Sum_{m>=0} q^(m^3)}^6.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A173677.

Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.

A173681 Number of ways of writing n as a sum of 8 nonnegative cubes.

Original entry on oeis.org

1, 8, 28, 56, 70, 56, 28, 8, 9, 56, 168, 280, 280, 168, 56, 8, 28, 168, 420, 560, 420, 168, 28, 0, 56, 280, 560, 568, 336, 224, 280, 280, 238, 336, 428, 336, 406, 840, 1120, 840, 392, 224, 168, 224, 840, 1680, 1680, 840, 196, 56, 28, 280, 1120, 1680, 1148, 448, 428, 568, 420, 448, 868, 840, 448, 840, 1689, 1736, 1008, 616, 616, 336, 476, 1688, 2576

Offset: 0

N. J. A. Sloane, Nov 25 2010

nonn

Order matters. This is the coefficient of q^n in the expansion of {Sum_{m>=0} q^(m^3)}^8.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
L. E. Dickson, All integers except 23 and 239 are sums of eight cubes, Bull. Amer. Math. Soc. 45 (1939), 588-591.

Cf. A004830, A173677.

Sums of k cubes, number of ways of writing n as, for k=1..9: A010057, A173677, A051343, A173678, A173679, A173680, A173676, A173681, A173682.

lista(n)=my(q='q); Vec(sum(m=0, (n+.5)^(1/3), q^(m^3), O(q^(n+1)))^8); \\ Michel Marcus, Apr 12 2016

cp's OEIS Frontend

A013675 Decimal expansion of zeta(17).

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A051343 Number of ways of writing n as a sum of 3 nonnegative cubes (counted naively).

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A113061 Sum of cube divisors of n.

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A173677 Number of ways of writing n as a sum of two nonnegative cubes.

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A173682 Number of ways of writing n as a sum of 9 nonnegative cubes.

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A173676 Number of ways of writing n as a sum of seven nonnegative cubes.

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A173678 Number of ways of writing n as a sum of 4 nonnegative cubes.

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A173679 Number of ways of writing n as a sum of 5 nonnegative cubes.

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A173680 Number of ways of writing n as a sum of 6 nonnegative cubes.

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A173681 Number of ways of writing n as a sum of 8 nonnegative cubes.