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.

14, 20, 42, 45, 52, 55, 121, 377, 451, 4331, 9211, 18042, 52093, 58743, 112853, 143498, 170322, 182293, 226118, 289811, 298489, 394067, 400831, 456745, 507678, 541362, 769667, 929582, 1132069, 1150595, 1218361, 1230403, 1262507, 1431109, 1516082, 1875155

Offset: 1

Author

Paolo P. Lava, Jul 08 2015

Keywords

• nonn

Examples

			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.
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.
		

Crossrefs

Cf. A001065, A066218, A066417.

Programs

Maple

with(numtheory): P:=proc(q) local a,b,i,j,k,n;
for n from 3 to q do a:=sort([op(divisors(n))]); b:=0;
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;
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;
if sigma(n)-n=b then print(n); fi; od; end: P(10^9);