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A342419 Sum of A342002 and its Dirichlet inverse.

%I A342419 #11 Mar 14 2021 20:42:58
%S A342419 2,0,0,1,0,10,0,3,25,14,0,7,0,14,70,15,0,-3,0,33,70,82,0,45,49,18,185,
%T A342419 29,0,8,0,35,410,94,98,28,0,22,90,97,0,0,0,79,279,106,0,-27,49,61,470,
%U A342419 43,0,-315,574,111,110,118,0,-199,0,18,339,43,126,-876,0,81,530,152,0,6,0,118,-87,153,574,412,0,267
%N A342419 Sum of A342002 and its Dirichlet inverse.
%C A342419 Zeros occurring on composite n are rare: 42 and 4718 are the first two such positions.
%C A342419 It seems that many nonzero squares occur on square n.
%H A342419 Antti Karttunen, <a href="/A342419/b342419.txt">Table of n, a(n) for n = 1..11550</a>
%F A342419 a(n) = A342002(n) + A342417(n).
%o A342419 (PARI)
%o A342419 up_to = 11550;
%o A342419 DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
%o A342419 A342002(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= p^(e>0); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
%o A342419 v342417 = DirInverseCorrect(vector(up_to,n,A342002(n)));
%o A342419 A342417(n) = v342417[n];
%o A342419 A342419(n) = (A342002(n)+A342417(n));
%Y A342419 Cf. A342002, A342417.
%K A342419 sign
%O A342419 1,1
%A A342419 _Antti Karttunen_, Mar 13 2021