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A329372 Dirichlet convolution of the identity function with A156552.

%I A329372 #9 Nov 12 2019 19:20:45
%S A329372 0,1,2,5,4,12,8,17,12,22,16,44,32,40,32,49,64,61,128,78,56,76,256,132,
%T A329372 32,142,50,136,512,152,1024,129,104,274,88,209,2048,532,188,230,4096,
%U A329372 256,8192,252,148,1048,16384,356,80,159,356,454,32768,240,160,392,680,2078,65536,504,131072,4128,248,321,280,464,262144,858,1328,400
%N A329372 Dirichlet convolution of the identity function with A156552.
%C A329372 Equally, Dirichlet convolution of sigma (A000203) with A297112 (Möbius transform of A156552).
%H A329372 Antti Karttunen, <a href="/A329372/b329372.txt">Table of n, a(n) for n = 1..1024</a>
%H A329372 Antti Karttunen, <a href="/A329372/a329372.txt">Data supplement: n, a(n) computed for n = 1..16384</a>
%H A329372 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A329372 a(n) = Sum_{d|n} d * A156552(n/d).
%F A329372 a(n) = Sum_{d|n} A000203(n/d) * A297112(d).
%F A329372 A000265(a(n)) = A329374(n).
%o A329372 (PARI)
%o A329372 A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
%o A329372 A329372(n) = sumdiv(n,d,(n/d)*A156552(d));
%o A329372 (PARI)
%o A329372 A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
%o A329372 A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
%o A329372 A297112(n) = if(1==n,0,2^A297167(n));
%o A329372 A329372(n) = sumdiv(n,d,sigma(n/d)*A297112(d));
%Y A329372 Cf. A000203, A061395, A156552, A297112, A297167, A329374.
%Y A329372 Cf. also A329371, A329373.
%K A329372 nonn
%O A329372 1,3
%A A329372 _Antti Karttunen_, Nov 12 2019