cp's OEIS Frontend

A259536 Numbers n such that sigma(n) - n = Sum_{k|n, k < n} sigma*(k), where sigma(n) is the sum of the divisors of n and sigma*(k) is the sum of the anti-divisors of k.

%I A259536 #47 Feb 14 2021 08:05:41
%S A259536 14,20,42,45,52,55,121,377,451,4331,9211,18042,52093,58743,112853,
%T A259536 143498,170322,182293,226118,289811,298489,394067,400831,456745,
%U A259536 507678,541362,769667,929582,1132069,1150595,1218361,1230403,1262507,1431109,1516082,1875155
%N A259536 Numbers n such that sigma(n) - n = Sum_{k|n, k < n} sigma*(k), where sigma(n) is the sum of the divisors of n and sigma*(k) is the sum of the anti-divisors of k.
%e A259536 Aliquot parts of 14 are 1, 2, 7. Anti-divisors of 7 are [2, 3, 5] while 1 and 2 have no anti-divisors. Finally 1 + 2 + 7 = 2 + 3 + 5 = 10.
%e A259536 Aliquot parts of 20 are 1, 2, 4, 5, 10. Anti-divisors: for 4 is [3], for 5 are [2, 3], for 10 are [3, 4, 7] while 1 and 2 have no anti-divisors. Finally 1 + 2 + 4 + 5 + 10 = 3 + 2 + 3 + 3 + 4 + 7 = 22.
%p A259536 with(numtheory): P:=proc(q) local a,b,i,j,k,n;
%p A259536 for n from 3 to q do a:=sort([op(divisors(n))]); b:=0;
%p A259536 for k from 2 to nops(a)-1 do i:=0; j:=a[k]; while j mod 2 <> 1 do i:=i+1; j:=j/2; od;
%p A259536 b:=b+sigma(2*a[k]+1)+sigma(2*a[k]-1)+sigma(a[k]/2^i)*2^(i+1)-6*a[k]-2; od;
%p A259536 if sigma(n)-n=b then print(n); fi; od; end: P(10^9);
%Y A259536 Cf. A001065, A066218, A066417.
%K A259536 nonn
%O A259536 1,1
%A A259536 _Paolo P. Lava_, Jul 08 2015