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%I #24 Oct 22 2022 08:06:33
%S 1,2,5,3,7,10,9,4,18,14,13,15,15,18,35,5,19,36,21,21,45,26,25,20,38,
%T 30,58,27,31,70,33,6,65,38,63,54,39,42,75,28,43,90,45,39,126,50,49,25,
%U 66,76,95,45,55,116,91,36,105,62,61,105,63,66,162,7,105,130,69
%N a(n) = Sum_{d|n} A000593(n/d).
%C Multiplicative because this sequence is the Dirichlet convolution of A000012 and A000593 which are both multiplicative. - _Andrew Howroyd_, Jul 27 2018
%H Seiichi Manyama, <a href="/A288417/b288417.txt">Table of n, a(n) for n = 1..10000</a>
%F L.g.f.: log(Product_{k>=1} (1 + x^k)^(sigma(k)/k)) = Sum_{n>=1} a(n)*x^n/n. - _Ilya Gutkovskiy_, Jun 19 2018
%F Multiplicative with a(2^e) = e+1 and a(p^e) = Sum_{i=0..e} (i+1)*p^(e-i) for e >= 0 and prime p > 2. - _Werner Schulte_, Jan 05 2021
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/144 = 0.676452... . - _Amiram Eldar_, Oct 22 2022
%t f[p_, e_] := Sum[(i + 1)*p^(e - i), {i, 0, e}]; f[2, e_] := e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 22 2022 *)
%o (PARI) a(n)={sumdiv(n, d, sigma(d>>valuation(d,2)))} \\ _Andrew Howroyd_, Jul 27 2018
%Y Cf. A000012.
%Y Sum_{d|n} d^k*A000593(n/d): this sequence (k=0), A109386 (k=1), A288418 (k=2), A288419 (k=3), A288420 (k=4).
%K nonn,mult
%O 1,2
%A _Seiichi Manyama_, Jun 09 2017