This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A132585 #21 Jul 27 2025 15:37:20 %S A132585 25,49,799,899,32399,292681,1492995736325809 %N 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. %C A132585 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? %C A132585 a(7) <= 1492995736325809. [From _Donovan Johnson_, Aug 31 2008] %C A132585 a(7) > 10^13. - _Giovanni Resta_, Jul 11 2013 %C A132585 No other terms < 2.7*10^15. - _Jud McCranie_, Jul 26 2025 %e A132585 k=25 -> sigma(25)= 1+5+25 -> sigma(k)-k-1=5 %e A132585 k+1=26 -> sigma(26)= 1+2+13+26 -> sigma(k+1)-k-2=2+13=15 %e A132585 15/5 = 3 (integer > 0) %p A132585 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); %Y A132585 Cf. A000203, A002961, A058072, A058073, A132586. %K A132585 nonn,hard,more %O A132585 1,1 %A A132585 _Paolo P. Lava_ and _Giorgio Balzarotti_, Aug 23 2007 %E A132585 a(6) from _Donovan Johnson_, Aug 31 2008 %E A132585 a(7) by _Jud McCranie_, Jul 26 2025