A132585 Numbers k such that sigma(k)-k-1 divides sigma(k+1)-k-2, where sigma(k) is sum of positive divisors of k and the ratio is greater than zero.

25, 49, 799, 899, 32399, 292681, 1492995736325809

Offset

1,1

Comments

The banal case of ratio equal to zero is excluded. In fact if k+1 is a prime than sigma(k+1)-k-2=0. Therefore the ratio with sigma(k)-k-1 is equal to zero. Is this sequence finite?

a(7) <= 1492995736325809. [From Donovan Johnson, Aug 31 2008]

a(7) > 10^13. - Giovanni Resta, Jul 11 2013

No other terms < 2.7*10^15. - Jud McCranie, Jul 26 2025

Links

Table of n, a(n) for n=1..7.

Example

k=25 -> sigma(25)= 1+5+25 -> sigma(k)-k-1=5
k+1=26 -> sigma(26)= 1+2+13+26 -> sigma(k+1)-k-2=2+13=15
15/5 = 3 (integer > 0)

Maple

with(numtheory); P:=proc(n) local a, i; for i from 1 by 1 to n do if sigma(i)-i-1>0 then a:=(sigma(i+1)-i-2)/(sigma(i)-i-1); if a>0 and trunc(a)=a then print(i); fi; fi; od; end: P(100000);

Crossrefs

Cf. A000203, A002961, A058072, A058073, A132586.

Sequence in context: A171590 A322434 A056981 * A049228 A309828 A076637

Adjacent sequences: A132582 A132583 A132584 * A132586 A132587 A132588

Keyword

nonn,hard,more

Author

Paolo P. Lava and Giorgio Balzarotti, Aug 23 2007

Extensions

a(6) from Donovan Johnson, Aug 31 2008

a(7) by Jud McCranie, Jul 26 2025

Status

approved