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A034778 Dirichlet convolution of Ramanujan numbers (A000594) with themselves.

%I A034778 #24 Dec 15 2017 07:27:26
%S A034778 1,-48,504,-2368,9660,-24192,-33488,239616,-163782,-463680,1069224,
%T A034778 -1193472,-1155476,1607424,4868640,86016,-13811868,7861536,21322840,
%U A034778 -22874880,-16877952,-51322752,37286544,120766464,-27669550,55462848,-203834232
%N A034778 Dirichlet convolution of Ramanujan numbers (A000594) with themselves.
%C A034778 Multiplicative because A000594 is. - _Christian G. Bower_, May 16 2005
%H A034778 G. C. Greubel, <a href="/A034778/b034778.txt">Table of n, a(n) for n = 1..10000</a>
%F A034778 a(n) = Sum_{d|n} tau(d)tau(n/d) where tau(n) = A000594(n) is Ramanujan's tau function.
%e A034778 G.f. = x - 48*x^2 + 504*x^3 - 2368*x^4 + 9660*x^5 - 24192*x^6 - 33488*x^7 + ...
%t A034778 a[n_] := DivisorSum[n, RamanujanTau[#]*RamanujanTau[n/#]&]; Array[a, 30] (* _Jean-François Alcover_, Nov 14 2015 *)
%o A034778 (PARI) {a(n) = local(A); if( n<1, 0, A = Vec( eta(x + x^n*O(x))^24); sumdiv(n, d, A[d] * A[n/d]))}; /* _Michael Somos_, Jul 16 2004 */
%o A034778 (Perl) use ntheory ":all"; for my $n (1..50) { say divisor_sum($n, sub { my $d=shift; ramanujan_tau($d)*ramanujan_tau($n/$d) } # _Dana Jacobsen_, Sep 05 2015
%Y A034778 Cf. A000594.
%K A034778 sign,mult
%O A034778 1,2
%A A034778 _N. J. A. Sloane_