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A309153 a(n) = A000203(n)*A001227(n).

%I A309153 #38 Dec 04 2023 01:33:36
%S A309153 1,3,8,7,12,24,16,15,39,36,24,56,28,48,96,31,36,117,40,84,128,72,48,
%T A309153 120,93,84,160,112,60,288,64,63,192,108,192,273,76,120,224,180,84,384,
%U A309153 88,168,468,144,96,248,171,279,288,196,108,480,288,240,320,180,120,672,124,192,624,127,336,576,136,252,384
%N A309153 a(n) = A000203(n)*A001227(n).
%C A309153 A001227(n) is denoted by Delta_0(n) in Glaisher 1907.
%C A309153 a(n) = A000203(n) iff n is a power of 2.
%H A309153 Amiram Eldar, <a href="/A309153/b309153.txt">Table of n, a(n) for n = 1..10000</a>
%H A309153 J. W. L. Glaisher, <a href="https://books.google.com/books?id=bLs9AQAAMAAJ&amp;pg=RA1-PA1">On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares</a>, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
%H A309153 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F A309153 a(n) = sigma(n)*delta(n).
%F A309153 Multiplicative with a(2^e) = 2^(e+1) - 1 and a(p^e) = (e+1)*(p^(e+1)-1)/(p-1) for p > 2. - _Amiram Eldar_, Nov 01 2022
%F A309153 From _Amiram Eldar_, Dec 04 2023: (Start)
%F A309153 Dirichlet g.f.: (4^s - 3*2^s + 2)/(4^s - 2) * (zeta(s)*zeta(s-1))^2/zeta(2*s-1).
%F A309153 Sum_{k=1..n} a(k) ~ (Pi^4/(168*zeta(3))) * n^2 * (log(n) + 2*gamma - 1/2 + 22*log(2)/21 + 2*zeta'(2)/zeta(2) - 2*zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620). (End)
%t A309153 Array[DivisorSum[#, 1 &, OddQ] DivisorSigma[1, #] &, 69] (* _Michael De Vlieger_, Nov 22 2019 *)
%t A309153 f[p_, e_] := (e+1)*(p^(e+1)-1)/(p-1); f[2, e_] := 2^(e+1) - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 01 2022 *)
%Y A309153 Cf. A000079, A000203, A001227, A062354, A062355, A064840.
%Y A309153 Cf. A001620, A002117, A244115, A306016
%K A309153 nonn,mult,easy
%O A309153 1,2
%A A309153 _Omar E. Pol_, Jul 14 2019