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A193553 Sum of divisors of 4*n.

%I A193553 #31 Sep 29 2024 21:10:53
%S A193553 7,15,28,31,42,60,56,63,91,90,84,124,98,120,168,127,126,195,140,186,
%T A193553 224,180,168,252,217,210,280,248,210,360,224,255,336,270,336,403,266,
%U A193553 300,392,378,294,480,308,372,546,360,336,508,399,465,504,434,378,600,504,504,560,450,420,744,434,480,728,511,588,720
%N A193553 Sum of divisors of 4*n.
%H A193553 Seiichi Manyama, <a href="/A193553/b193553.txt">Table of n, a(n) for n = 1..10000</a>
%F A193553 a(n) = sigma(4*n) = A000203(4*n).
%F A193553 a(n) = 3*sigma(2*n) - 2*sigma(n); the relation is the special case e=1, p=2 of the relation sigma(t^2*n) = (t+1)*sigma(t*n) - t*sigma(n) where t=p^e (p a prime).
%F A193553 G.f. is x times the logarithmic derivative of the g.f. of A182820.
%F A193553 a(2*n-1) = 7 * A008438(n) = 7 * sigma(2*n-1); special case of sigma(2^k*(2*n-1)) = (2^(k+1)-1) * sigma(2*n-1).
%F A193553 Sum_{k=1..n} a(k) = (11*Pi^2/24) * n^2 + O(n*log(n)). - _Amiram Eldar_, Dec 16 2022
%F A193553 G.f.: Sum_{k>=1} k*x^(k/gcd(k, 4))/(1 - x^(k/gcd(k, 4))). - _Miles Wilson_, Sep 29 2024
%t A193553 DivisorSigma[1,4*Range[70]] (* _Harvey P. Dale_, Jan 27 2015 *)
%o A193553 (PARI) vector(66, n, sigma(4*n, 1))
%Y A193553 Sigma(k*n): A000203 (k=1), A062731 (k=2), A144613 (k=3), this sequence (k=4), A283118 (k=5), A224613 (k=6), A283078 (k=7), A283122 (k=8), A283123 (k=9).
%Y A193553 Cf. A008438, A008586, A182820.
%K A193553 nonn
%O A193553 1,1
%A A193553 _Joerg Arndt_, Jul 30 2011