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A349438 Dirichlet convolution of A000027 with 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 A349438 #16 Nov 21 2021 01:19:22
%S A349438 1,1,1,1,2,0,2,1,3,0,4,-1,2,0,2,1,4,-1,2,-2,2,0,4,-2,10,0,9,-2,6,-4,2,
%T A349438 1,4,0,4,-4,6,0,2,-4,4,-4,2,-4,6,0,4,-3,14,-4,4,-2,6,-6,8,-4,2,0,6,-6,
%U A349438 2,0,6,1,4,-8,6,-4,4,-8,4,-6,2,0,10,-2,8,-4,6,-6,27,0,4,-6,8,0,6,-8,6,-16,4,-4,2,0,4
%N A349438 Dirichlet convolution of A000027 with 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.
%C A349438 Convolving this sequence with A348045 gives Euler phi, A000010.
%C A349438 It might first seem that A000265(a(p^k)) = p^(k-1) for all odd primes and all exponents k >= 1, but this does not hold for prime 37. However, with p=37, identity A065330(A349438(37^k)) = 37^(k-1) seems to hold for all exponents k >= 1. - _Antti Karttunen_, Nov 20 2021
%H A349438 Antti Karttunen, <a href="/A349438/b349438.txt">Table of n, a(n) for n = 1..20000</a>
%H A349438 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A349438 a(n) = Sum_{d|n} d * A349348(n/d).
%t A349438 f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := If[EvenQ[n], n/2, Times @@ f @@@ FactorInteger[n]]; sinv[1] = 1; sinv[n_] := sinv[n] = -DivisorSum[n, sinv[#] * s[n/#] &, # < n &]; a[n_] := DivisorSum[n, # * sinv[n/#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 18 2021 *)
%o A349438 (PARI)
%o A349438 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 A349438 A252463(n) = if(!(n%2),n/2,A064989(n));
%o A349438 memoA349348 = Map();
%o A349438 A349348(n) = if(1==n,1,my(v); if(mapisdefined(memoA349348,n,&v), v, v = -sumdiv(n,d,if(d<n,A252463(n/d)*A349348(d),0)); mapput(memoA349348,n,v); (v)));
%o A349438 A349438(n) = sumdiv(n,d,d*A349348(n/d));
%Y A349438 Cf. A000027, A064989, A252463, A349348, A349437 (Dirichlet inverse), A349439 (sum with it).
%Y A349438 Cf.  also A000010, A000265, A065330, A348045.
%K A349438 sign
%O A349438 1,5
%A A349438 _Antti Karttunen_, Nov 18 2021