A132630 Numbers n such that sigma(n)-n divides sigma(n+1)-n-1, where sigma(n) is sum of positive divisors of n.
2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 85, 89, 97, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199
Offset: 1
Examples
n=85 -> sigma(n+1)-n-1=1+2+43=46 sigma(n)-n=1+5+17=23 -> 46/23=2 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
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
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Magma
[k:k in [2..200]| IsIntegral((DivisorSigma(1,k+1)-k-1)/ (DivisorSigma(1,k)-k))]; // Marius A. Burtea, Nov 06 2019
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Maple
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)
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Mathematica
Select[Range[2,200],Divisible[DivisorSigma[1,#+1]-#-1,DivisorSigma[ 1,#]-#]&] (* Harvey P. Dale, Apr 25 2015 *)
Extensions
Comment amended by Harvey P. Dale, Apr 25 2015
Comments