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

Author

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

Keywords

• nonn

Comments

a(n) is a perfect number for n = 5 and n = r*q with r = 4*46817 and q = 4477417228433 = (A006516(31)-sigma(r))/a(r) [Radcliffe, 2025]. Are there other n with this property? - M. F. Hasler, Mar 10 2025

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• David Radcliffe, in reply to Leo Hennig, Re: One entry and just one entry, SeqFan list, March 6, 2025.

Crossrefs

Cf. A000203, A001065, A013661.

Programs

Mathematica

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

PARI

a(n) = sigma(n) + n; /* Michael Somos, Sep 19 2011 */

Formula

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.

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

Extensions

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

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