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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.

%I A224226 #13 Aug 17 2019 15:43:17
%S A224226 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,
%T A224226 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,
%U A224226 -24,20,0,0,6,0,0,32,-31,0,24,0,-18,24,0,0,5,0,0,31,-20,0,28
%N 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.
%H A224226 Amiram Eldar, <a href="/A224226/b224226.txt">Table of n, a(n) for n = 0..10000</a>
%H A224226 K. S. Williams, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.120.04.329">The parents of Jacobi's four squares theorem are unique</a>, Amer. Math. Monthly, 120 (2013), 329-345.
%p A224226 s := proc(n,k)
%p A224226     if modp(n,k) = 0 then
%p A224226         numtheory[sigma](n/k) ;
%p A224226     else
%p A224226         0 ;
%p A224226     end if;
%p A224226 end proc:
%p A224226 A224226 := proc(n)
%p A224226     if n = 0 then
%p A224226         1;
%p A224226     else
%p A224226         s(n,3)-s(n,4)-s(n,6)+s(n,12) ;
%p A224226     end if;
%p A224226 end proc: # _R. J. Mathar_, Nov 14 2018
%t A224226 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 *)
%o A224226 (PARI) s(n,k) = if (!(n%k), sigma(n/k), 0);
%o A224226 a(n) = if (n==0, 1, s(n,3)-s(n,4)-s(n,6)+s(n,12)); \\ _Michel Marcus_, Sep 27 2017
%Y A224226 Cf. A000203, A321527.
%K A224226 sign
%O A224226 0,7
%A A224226 _N. J. A. Sloane_, Apr 09 2013