|
| |
|
|
A155085
|
|
a(n) = n + sum of divisors of n.
|
|
11
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
|
|
|
|
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.
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
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 19 2009, Jan 25 2009, Jan 26 2009
|
|
|
EXTENSIONS
|
Zero removed and offset corrected by Omar E. Pol, Jan 27 2009
|
|
|
STATUS
|
approved
|
| |
|
|
