A193553 Sum of divisors of 4*n.

7, 15, 28, 31, 42, 60, 56, 63, 91, 90, 84, 124, 98, 120, 168, 127, 126, 195, 140, 186, 224, 180, 168, 252, 217, 210, 280, 248, 210, 360, 224, 255, 336, 270, 336, 403, 266, 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

Offset: 1

Author

Joerg Arndt, Jul 30 2011

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

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).

Cf. A008438, A008586, A182820.

Programs

Mathematica

DivisorSigma[1,4*Range[70]] (* Harvey P. Dale, Jan 27 2015 *)

PARI

vector(66, n, sigma(4*n, 1))

Formula

a(n) = sigma(4*n) = A000203(4*n).

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).

G.f. is x times the logarithmic derivative of the g.f. of A182820.

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).

Sum_{k=1..n} a(k) = (11*Pi^2/24) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 16 2022

G.f.: Sum_{k>=1} k*x^(k/gcd(k, 4))/(1 - x^(k/gcd(k, 4))). - Miles Wilson, Sep 29 2024