A132630 Numbers n such that sigma(n)-n divides sigma(n+1)-n-1, where sigma(n) is sum of positive divisors of n.

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 85, 89, 97, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199

Offset: 1

Author

Paolo P. Lava and Giorgio Balzarotti, Aug 27 2007

Keywords

• easy
• nonn

Comments

With the exception of the first term, only odd numbers. All the prime numbers p are included because sigma(p)-p=1.

Examples

			n=85 -> sigma(n+1)-n-1=1+2+43=46 sigma(n)-n=1+5+17=23 -> 46/23=2
n=125 -> sigma(n+1)-n-1=1+2+3+6+7+9+14+18+21+42+63=186 sigma(n)-n=1+5+25=31 -> 186/31=6
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A132585, A132586, A132631.

Programs

Magma

[k:k in [2..200]| IsIntegral((DivisorSigma(1,k+1)-k-1)/ (DivisorSigma(1,k)-k))]; // Marius A. Burtea, Nov 06 2019

Maple

with(numtheory); P:=proc(n) local a,i; for i from 1 by 1 to n do if sigma(i)-i>0 then a:=(sigma(i+1)-i-1)/(sigma(i)-i); if a>0 and trunc(a)=a then print(i); fi; fi; od; end: P(200)

Mathematica

Select[Range[2,200],Divisible[DivisorSigma[1,#+1]-#-1,DivisorSigma[ 1,#]-#]&] (* Harvey P. Dale, Apr 25 2015 *)

Extensions

Comment amended by Harvey P. Dale, Apr 25 2015