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

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

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.

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

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.

Mathematica

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

Prog

(PARI) a(n) = 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,changed

Author

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