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A349397 Dirichlet convolution of A064216 with the Dirichlet inverse of its inverse permutation.

%I A349397 #12 Nov 25 2021 20:30:06
%S A349397 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 A349397 -16,18,-24,40,-70,-9,24,-21,8,50,-55,-8,24,-6,31,15,-58,-20,17,31,
%U A349397 -92,-2,-70,37,24,0,20,49,18,-6,-26,13,-33,15,-62,-158,-20,22,-15,49,-130,67,48,49,-58,29,-112,-4,60,-73,-16
%N A349397 Dirichlet convolution of A064216 with the Dirichlet inverse of its inverse permutation.
%C A349397 Dirichlet convolution of A064216 with A323893, which is the Dirichlet inverse of A048673. Therefore, convolving A048673 with this sequence gives A064216.
%C A349397 Note how for n = 1 .. 35, a(n) = -A349398(n).
%H A349397 Antti Karttunen, <a href="/A349397/b349397.txt">Table of n, a(n) for n = 1..16384</a>
%H A349397 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A349397 a(n) = Sum_{d|n} A064216(n/d) * A323893(d).
%o A349397 (PARI)
%o A349397 A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(1+factorback(f)); };
%o A349397 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 A349397 memoA323893 = Map();
%o A349397 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 A349397 A349397(n) = sumdiv(n,d,A064216(n/d)*A323893(d));
%Y A349397 Cf. A003961, A048673, A064216, A064989, A323893, A349398 (Dirichlet inverse), A349399 (sum with it), A349384.
%Y A349397 Cf. also pairs A349376, A349377 and A349613, A349614 for similar constructions.
%K A349397 sign
%O A349397 1,7
%A A349397 _Antti Karttunen_, Nov 19 2021