A224226 a(0)=1; thereafter a(n) =s(n,3)-s(n,4)-s(n,6)+s(n,12), where s(n,k) = sigma(n/k) if k divides n, otherwise 0.

1, 0, 0, 1, -1, 0, 2, 0, -3, 4, 0, 0, 1, 0, 0, 6, -7, 0, 8, 0, -6, 8, 0, 0, -1, 0, 0, 13, -8, 0, 12, 0, -15, 12, 0, 0, 7, 0, 0, 14, -18, 0, 16, 0, -12, 24, 0, 0, -5, 0, 0, 18, -14, 0, 26, 0, -24, 20, 0, 0, 6, 0, 0, 32, -31, 0, 24, 0, -18, 24, 0, 0, 5, 0, 0, 31, -20, 0, 28

Offset: 0

Author

N. J. A. Sloane, Apr 09 2013

Keywords

• sign

Links

• Amiram Eldar, Table of n, a(n) for n = 0..10000
• K. S. Williams, The parents of Jacobi's four squares theorem are unique, Amer. Math. Monthly, 120 (2013), 329-345.

Crossrefs

Cf. A000203, A321527.

Programs

Maple

s := proc(n,k)
    if modp(n,k) = 0 then
        numtheory[sigma](n/k) ;
    else
        0 ;
    end if;
end proc:
A224226 := proc(n)
    if n = 0 then
        1;
    else
        s(n,3)-s(n,4)-s(n,6)+s(n,12) ;
    end if;
end proc: # R. J. Mathar, Nov 14 2018

Mathematica

s[n_, k_] := If[Divisible[n, k], DivisorSigma[1, n/k], 0]; a[0] = 1; a[n_] := s[n, 3] - s[n, 4] - s[n, 6] + s[n, 12]; Array[a, 100, 0] (* Amiram Eldar, Aug 17 2019 *)

PARI

s(n,k) = if (!(n%k), sigma(n/k), 0);
a(n) = if (n==0, 1, s(n,3)-s(n,4)-s(n,6)+s(n,12)); \\ Michel Marcus, Sep 27 2017