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A349349 Sum of A252463 and its Dirichlet inverse, where A252463 shifts the prime factorization of odd numbers one step towards smaller primes and divides even numbers by two.

%I A349349 #17 Apr 21 2022 09:14:59
%S A349349 2,0,0,1,0,4,0,3,4,6,0,8,0,10,12,7,0,8,0,13,20,14,0,15,9,22,8,19,0,14,
%T A349349 0,15,28,26,30,19,0,34,44,25,0,18,0,29,12,38,0,28,25,21,52,37,0,24,42,
%U A349349 35,68,46,0,28,0,58,20,31,66,30,0,47,76,32,0,38,0,62,18,55,70,30,0,47,16,74,0,36,78,82,92,55
%N A349349 Sum of A252463 and its Dirichlet inverse, where A252463 shifts the prime factorization of odd numbers one step towards smaller primes and divides even numbers by two.
%C A349349 Question: Are there any negative terms? All terms in range 1 .. 2^23 are nonnegative. (See also A349126). - _Antti Karttunen_, Apr 20 2022
%H A349349 Antti Karttunen, <a href="/A349349/b349349.txt">Table of n, a(n) for n = 1..20000</a>
%H A349349 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A349349 a(n) = A252463(n) + A349348(n).
%F A349349 a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1<d<n} A252463(d) * A349348(n/d).
%F A349349 For all n >= 1, a(2n-1) = A349126(2n-1).
%o A349349 (PARI)
%o A349349 up_to = 20000;
%o A349349 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 A349349 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 A349349 A252463(n) = if(!(n%2),n/2,A064989(n));
%o A349349 v349348 = DirInverseCorrect(vector(up_to,n,A252463(n)));
%o A349349 A349348(n) = v349348[n];
%o A349349 A349349(n) = (A252463(n)+A349348(n));
%Y A349349 Coincides with A349126 on odd numbers.
%Y A349349 Cf. A064989, A252463, A349348.
%Y A349349 Cf. also A323365, A323412, A323894, A349135, A353336.
%K A349349 nonn
%O A349349 1,1
%A A349349 _Antti Karttunen_, Nov 15 2021