A050456 a(n) = Sum_{d|n, d==1 mod 4} d^4 - Sum_{d|n, d==3 mod 4} d^4.

1, 1, -80, 1, 626, -80, -2400, 1, 6481, 626, -14640, -80, 28562, -2400, -50080, 1, 83522, 6481, -130320, 626, 192000, -14640, -279840, -80, 391251, 28562, -524960, -2400, 707282, -50080, -923520, 1, 1171200, 83522, -1502400, 6481, 1874162, -130320

Offset

1,3

Comments

Multiplicative because it is the Inverse Möbius transform of [1 0 -3^4 0 5^4 0 -7^4 ...], which is multiplicative. - Christian G. Bower, May 18 2005

Called E_4(n) by Hardy.

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

References

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 120.

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Chelsea Publishing Company, New York 1959, p. 135 section 9.3. MR0106147 (21 #4881)

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).

Ken Ono, Sinai Robins, and Patrick T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, Volume 50, Issue 1-2 (August 1995), pp. 73-94. Case k=10; Author's copy. [Wayback Machine link]

Michael Somos, Introduction to Ramanujan theta functions, 2019.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Index entries for sequences mentioned by Glaisher.

Formula

Expansion of ( eta(q)^4 * eta(q^2)^2 * (5 * eta(q)^8 / eta(q^4)^4 + 64 * q * eta(q^4)^4 ) - 5) / 4 in powers of q. - Michael Somos, Jan 14 2012

Expansion of (phi(x)^2 * (5 * phi(-x)^8 + 64 * x * psi(-x)^8) - 5) / 4 in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Jan 14 2012

a(n) is multiplicative with a(2^e) = 1, a(p^e) = ((p^4)^(e+1) - 1) / (p^4 - 1) if p == 1 (mod 4), a(p^e) = (1 - (-p^4)^(e+1)) / (1 + p^4) if p == 3 (mod 4). - Michael Somos, Jan 14 2012

G.f.: Sum_{k>0} (-1)^(k-1) * (2*k - 1)^4 * x^(2*k - 1) / (1 - x^(2*k - 1)).

a(n) = Sum_{d|n} Chi(d)*d^4, with Chi(n) the second Dirichlet character modulo 4, i.e., Chi(1) = 1 and Chi(3) = -1. See the Ono et al ref. case k=10. - Wolfdieter Lang, Jan 13 2017

Example

G.f. = x + x^2 - 80*x^3 + x^4 + 626*x^5 - 80*x^6 - 2400*x^7 + x^8 + 6481*x^9 + ...

Mathematica

e[r_, n_] := Plus@@(Select[Divisors[n], Mod[#, 4] == 1 &]^r) -Plus@@(Select[Divisors[n], Mod[#, 4] == 3 &]^r); e[4, #] &/@Range[38] (* Ant King, Nov 10 2012 *)
f[p_, e_] := If[Mod[p, 4] == 1, ((p^4)^(e+1)-1)/(p^4-1), ((-p^4)^(e+1)-1)/(-p^4-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Sep 27 2023 *)

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (d%2) * (-1)^((d-1)/2) * d^4))}; /* Michael Somos, Sep 12 2005 */
(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, d^4 * kronecker( -4, d)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x + A)^4 * eta(x^2 + A)^2 * (5 * eta(x + A)^8 / eta(x^4 + A)^4 + 64 * x * eta(x^4 + A)^4 ) - 5) / 4, n))}; /* Michael Somos, Jan 14 2012 */

Crossrefs

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A093404 A236236 A031136 * A107930 A239131 A033400

Adjacent sequences: A050453 A050454 A050455 * A050457 A050458 A050459

Keyword

sign,easy,mult

Author

N. J. A. Sloane, Dec 23 1999

Status

approved