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A349348 Dirichlet inverse of A252463, where A252463 shifts the prime factorization of odd numbers one step towards smaller primes and divides even numbers by two.

%I A349348 #11 Nov 21 2021 01:18:07
%S A349348 1,-1,-2,-1,-3,1,-5,-1,0,1,-7,2,-11,3,6,-1,-13,-1,-17,3,10,3,-19,3,0,
%T A349348 9,0,5,-23,-1,-29,-1,14,9,15,1,-31,15,22,5,-37,-3,-41,7,0,15,-43,4,0,
%U A349348 -4,26,11,-47,-3,21,7,34,17,-53,-2,-59,27,0,-1,33,-3,-61,13,38,-3,-67,2,-71,25,0,17,35,-9,-73,7,0,33
%N A349348 Dirichlet inverse of A252463, where A252463 shifts the prime factorization of odd numbers one step towards smaller primes and divides even numbers by two.
%H A349348 Antti Karttunen, <a href="/A349348/b349348.txt">Table of n, a(n) for n = 1..20000</a>
%H A349348 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A349348 a(1) = 1; a(n) = -Sum_{d|n, d < n} A252463(n/d) * a(d).
%F A349348 a(n) = A349349(n) - A252463(n).
%F A349348 For all n >= 1, a(2n-1) = A349125(2n-1).
%o A349348 (PARI)
%o A349348 up_to = 20000;
%o A349348 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 A349348 A064989(n) = {my(f); f = factor(n); 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 A349348 A252463(n) = if(!(n%2),n/2,A064989(n));
%o A349348 v349348 = DirInverseCorrect(vector(up_to,n,A252463(n)));
%o A349348 A349348(n) = v349348[n];
%Y A349348 Coincides with A349125 on odd numbers.
%Y A349348 Cf. A064989, A252463, A349349.
%Y A349348 Cf. also A348045, A349437, A349438.
%K A349348 sign
%O A349348 1,3
%A A349348 _Antti Karttunen_, Nov 15 2021