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A349398 Dirichlet convolution of A048673 with the Dirichlet inverse of its inverse permutation.

%I A349398 #11 Nov 25 2021 20:30:30
%S A349398 1,0,0,0,0,1,-5,8,0,-6,-3,2,0,19,-5,-4,-4,20,-19,22,6,-15,3,-8,0,0,16,
%T A349398 16,-18,24,-40,70,9,-24,21,-7,-50,55,8,-24,6,-41,-15,58,20,-17,-31,
%U A349398 108,27,70,-37,-24,0,-20,-49,-98,6,26,-13,21,-15,62,158,84,-22,9,-49,130,-67,12,-49,62,-29,112,4,-60,103,16
%N A349398 Dirichlet convolution of A048673 with the Dirichlet inverse of its inverse permutation.
%C A349398 Dirichlet convolution of A048673 with A349358, which is the Dirichlet inverse of A064216 (inverse permutation of A048673). Therefore, convolving A064216 with this sequence gives A048673.
%C A349398 Note how for n = 1 .. 35, a(n) = -A349397(n).
%H A349398 Antti Karttunen, <a href="/A349398/b349398.txt">Table of n, a(n) for n = 1..16384</a>
%H A349398 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A349398 a(n) = Sum_{d|n} A048673(n/d) * A349358(d).
%o A349398 (PARI)
%o A349398 A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(1+factorback(f)); };
%o A349398 A064216(n) = { my(f = factor(n+n-1)); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
%o A349398 memoA349358 = Map();
%o A349398 A349358(n) = if(1==n,1,my(v); if(mapisdefined(memoA349358,n,&v), v, v = -sumdiv(n,d,if(d<n,A064216(n/d)*A349358(d),0)); mapput(memoA349358,n,v); (v)));
%o A349398 A349398(n) = sumdiv(n,d,A048673(n/d)*A349358(d));
%Y A349398 Cf. A003961, A048673, A064216, A064989, A323893, A349397 (Dirichlet inverse), A349399 (sum with it).
%Y A349398 Cf. also A349376, A349377, A349385.
%K A349398 sign
%O A349398 1,7
%A A349398 _Antti Karttunen_, Nov 19 2021