A155085 a(n) = n + sum of divisors of n.

2, 5, 7, 11, 11, 18, 15, 23, 22, 28, 23, 40, 27, 38, 39, 47, 35, 57, 39, 62, 53, 58, 47, 84, 56, 68, 67, 84, 59, 102, 63, 95, 81, 88, 83, 127, 75, 98, 95, 130, 83, 138, 87, 128, 123, 118, 95, 172, 106, 143, 123, 150, 107, 174, 127, 176, 137, 148, 119, 228, 123, 158, 167, 191

Offset

1,1

Comments

If n is a prime, a(n)=2n+1, if n is a perfect number a(n)=3n, if n is of the form 2^m, a(n) = 2^(m+1) + 2^m -1. Generally a(n)>=2n+1.

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Formula

If n = Product_{i=1...k} Pi^ki is the prime power factorization of n, then a(n) = n + [Product_{i=1...k} {Pi^(ki+1)-1}]/[Product_{i=1...k} (Pi-1)]. For example, 18 = 2*3^2; a(18) = 18+(2^2-1)*(3^3-1)/[(2-1)*(3-1)] = 57.

a(n) = A000203(n) + n = A001065(n) + 2*n. - Michael Somos, Sep 19 2011

a(n) = A001065(-n). - Michael Somos, Sep 20 2011

G.f.: 1/(1-x)+1/(1-x)^2 - 2*x/(Q(0) - 2*x^2 + 2*x), where Q(k)= (2*x^(k+2) - x - 1)*k - 1 - 2*x + 3*x^(k+2) - x*(k+3)*(k+1)*(1-x^(k+2))^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 16 2013

G.f.: x/(1 - x)^2 + Sum_{k>=1} x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Mar 17 2017

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = (zeta(2)+1)/2 = 1.322467... . - Amiram Eldar, Mar 17 2024

Example

Taking 18 as an example, a(18) = 18+1+2+3+6+9+18=57; 18=2*3^2; a(18)=18+(2^2-1)*(3^3-1)/[(2-1)*(3-1)]=57. [Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 25 2009]

Mathematica

Table[n+DivisorSigma[1, n], {n, 70}] (* Harvey P. Dale, Apr 27 2019 *)

Prog

(PARI) {a(n) = if( n==0, 0, sigma(n) + n)}; /* Michael Somos, Sep 19 2011 */

Crossrefs

Cf. A000203, A001065, A013661.

Sequence in context: A257325 A020484 A124203 * A077665 A025062 A360019

Adjacent sequences: A155082 A155083 A155084 * A155086 A155087 A155088

Keyword

nonn

Author

Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 19 2009, Jan 25 2009, Jan 26 2009

Extensions

More terms from N. J. A. Sloane, Jan 24 2009

Zero removed and offset corrected by Omar E. Pol, Jan 27 2009

Status

approved