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A349572 Dirichlet convolution of A000027 (identity function) with the Dirichlet inverse of A048673.

%I A349572 #15 Nov 26 2021 10:34:49
%S A349572 1,0,0,-1,1,-2,1,-4,-4,-3,4,-4,4,-5,-6,-12,7,-6,7,-7,-10,-6,8,-6,-4,
%T A349572 -8,-24,-11,13,-2,12,-32,-12,-9,-14,-4,16,-11,-16,-13,19,-2,19,-16,
%U A349572 -22,-14,20,-4,-18,-15,-18,-20,23,-10,-14,-19,-22,-15,28,14,27,-18,-34,-80,-20,-8,31,-25,-28,-8,34,14,33,-20
%N A349572 Dirichlet convolution of A000027 (identity function) with the Dirichlet inverse of A048673.
%C A349572 Also Dirichlet convolution of A349384 with A349388.
%H A349572 Antti Karttunen, <a href="/A349572/b349572.txt">Table of n, a(n) for n = 1..20000</a>
%H A349572 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A349572 a(n) = Sum_{d|n} d * A323893(n/d).
%F A349572 a(n) = Sum_{d|n} A349384(d) * A349388(n/d).
%t A349572 f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := (1 + Times @@ f @@@ FactorInteger[n])/2; 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 23 2021 *)
%o A349572 (PARI)
%o A349572 A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(1+factorback(f)); };
%o A349572 memoA323893 = Map();
%o A349572 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 A349572 A349572(n) = sumdiv(n,d,d*A323893(n/d));
%Y A349572 Cf. A000027, A048673, A323893, A349384, A349388, A349571 (Dirichlet inverse).
%Y A349572 Cf. also A349397.
%K A349572 sign
%O A349572 1,6
%A A349572 _Antti Karttunen_, Nov 23 2021