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%I #14 Sep 08 2022 08:45:31
%S 2,3,5,7,9,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,
%T 85,89,97,101,103,107,109,113,125,127,131,137,139,149,151,157,163,167,
%U 169,173,179,181,191,193,197,199
%N Numbers n such that sigma(n)-n divides sigma(n+1)-n-1, where sigma(n) is sum of positive divisors of n.
%C With the exception of the first term, only odd numbers. All the prime numbers p are included because sigma(p)-p=1.
%H Amiram Eldar, <a href="/A132630/b132630.txt">Table of n, a(n) for n = 1..10000</a>
%e n=85 -> sigma(n+1)-n-1=1+2+43=46 sigma(n)-n=1+5+17=23 -> 46/23=2
%e 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
%p 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)
%t Select[Range[2,200],Divisible[DivisorSigma[1,#+1]-#-1,DivisorSigma[ 1,#]-#]&] (* _Harvey P. Dale_, Apr 25 2015 *)
%o (Magma) [k:k in [2..200]| IsIntegral((DivisorSigma(1,k+1)-k-1)/ (DivisorSigma(1,k)-k))]; // _Marius A. Burtea_, Nov 06 2019
%Y Cf. A132585, A132586, A132631.
%K easy,nonn
%O 1,1
%A _Paolo P. Lava_ and _Giorgio Balzarotti_, Aug 27 2007
%E Comment amended by _Harvey P. Dale_, Apr 25 2015
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