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

1,1

Comments

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

Links

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A132585, A132586, A132631.

Sequence in context: A353393 A316857 A318589 * A033946 A328232 A189710

Adjacent sequences: A132627 A132628 A132629 * A132631 A132632 A132633

Keyword

easy,nonn

Author

Paolo P. Lava and Giorgio Balzarotti, Aug 27 2007

Extensions

Comment amended by Harvey P. Dale, Apr 25 2015

Status

approved