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

Author

Paolo P. Lava and Giorgio Balzarotti, Aug 23 2007

Keywords

• nonn
• hard
• more

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

Examples

			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)
		

Crossrefs

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

Programs

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);

Extensions

a(6) from Donovan Johnson, Aug 31 2008

a(7) by Jud McCranie, Jul 26 2025