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A323894 Sum of A048673 and its Dirichlet inverse, A323893.

%I A323894 #17 Nov 30 2024 08:50:27
%S A323894 2,0,0,4,0,12,0,12,9,16,0,26,0,24,24,37,0,46,0,36,36,28,0,76,16,36,51,
%T A323894 56,0,58,0,114,42,40,48,121,0,48,54,106,0,94,0,66,104,60,0,223,36,92,
%U A323894 60,86,0,220,56,166,72,64,0,164,0,76,162,349,72,112,0,96,90,136,0,354,0,84,150,116,84,148,0,312,277,88,0,260,80,96,96
%N A323894 Sum of A048673 and its Dirichlet inverse, A323893.
%C A323894 The first four negative terms are a(3063060) = -14126242, a(3423420) = -17546656, a(4084080) = -14460312, a(4144140) = -22677277. - _Antti Karttunen_, Apr 20 2022
%H A323894 Antti Karttunen, <a href="/A323894/b323894.txt">Table of n, a(n) for n = 1..16383</a>
%H A323894 Antti Karttunen, <a href="/A323894/a323894.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%H A323894 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%F A323894 a(n) = A048673(n) + A323893(n).
%F A323894 For n > 1, a(n) = -Sum_{d|n, 1<d<n} A048673(n/d) * A323893(d). - _Antti Karttunen_, Apr 20 2022
%F A323894 a(n) = A349135(A003961(n)). - _Antti Karttunen_, Nov 30 2024
%o A323894 (PARI)
%o A323894 up_to = 65537;
%o A323894 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 A323894 A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
%o A323894 A048673(n) = (A003961(n)+1)/2;
%o A323894 v323893 = DirInverse(vector(up_to,n,A048673(n)));
%o A323894 A323893(n) = v323893[n];
%o A323894 A323894(n) = (A048673(n)+A323893(n));
%Y A323894 Cf. A003961, A048673, A323893, A349135.
%Y A323894 Cf. also A323365, A323412, A323896, A349135, A349349, A353336.
%K A323894 sign
%O A323894 1,1
%A A323894 _Antti Karttunen_, Feb 08 2019