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A349384 Dirichlet convolution of A003961 with the Dirichlet inverse of A048673, where A003961 is fully multiplicative with a(p) = nextprime(p), and A048673(n) = (1+A003961(n))/2.

%I A349384 #11 Nov 26 2021 08:52:30
%S A349384 1,1,2,2,3,0,5,4,6,0,6,-2,8,0,0,8,9,-4,11,-3,0,0,14,-8,12,0,18,-5,15,
%T A349384 -12,18,16,0,0,0,-14,20,0,0,-12,21,-20,23,-6,-12,0,26,-24,30,-9,0,-8,
%U A349384 29,-24,0,-20,0,0,30,-24,33,0,-20,32,0,-24,35,-9,0,-30,36,-36,39,0,-18,-11,0,-32,41,-36,54,0,44
%N A349384 Dirichlet convolution of A003961 with the Dirichlet inverse of A048673, where A003961 is fully multiplicative with a(p) = nextprime(p), and A048673(n) = (1+A003961(n))/2.
%C A349384 Convolving this with A336840 gives A003973.
%H A349384 Antti Karttunen, <a href="/A349384/b349384.txt">Table of n, a(n) for n = 1..20000</a>
%H A349384 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A349384 a(n) = Sum_{d|n} A003961(n/d) * A323893(d).
%F A349384 a(n) = A349386(n) - A349385(n).
%o A349384 (PARI)
%o A349384 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A349384 A048673(n) = (A003961(n)+1)/2;
%o A349384 memoA323893 = Map();
%o A349384 A323893(n) = if(1==n,1,my(v); if(mapisdefined(memoA323893,n,&v), v, v = -sumdiv(n,d,if(d<n,A048673(n/d)*A323893(d),0)); mapput(memoA323893,n,v); (v)));
%o A349384 A349384(n) = sumdiv(n,d,A003961(n/d)*A323893(d));
%Y A349384 Cf. A003961, A048673, A323893, A349385 (Dirichlet inverse), A349386 (sum with it).
%Y A349384 Cf. also A003973, A336840, A349572.
%K A349384 sign
%O A349384 1,3
%A A349384 _Antti Karttunen_, Nov 17 2021