A132586 Numbers k such that sigma(k+1)-k-2 divides sigma(k)-k-1, where sigma(k) is sum of positive divisors of k and the ratio is greater than zero.

8, 24, 8925, 32445, 118540859325

Offset: 1

Author

Paolo P. Lava and Giorgio Balzarotti, Aug 23 2007

Keywords

• hard
• more
• nonn

Comments

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

a(6), if it exists, is larger than 10^13. - Giovanni Resta, Jul 13 2015

No more terms < 2.7*10^15. - Jud McCranie, Jul 27 2025

Examples

			n=8 -> sigma(8)=1+2+4+8 -> sigma(n)-n-1=2+4=6.
n+1=9 -> sigma(9)=1+3+9 -> sigma(n+1)-n-2=3.
6/3 = 2 (integer >0)
		

Crossrefs

Cf. A002961, A058072, A058073, A132585.

Programs

Maple

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

Extensions

a(5) from Donovan Johnson, Aug 31 2008