login
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.
0
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
1,1
EXAMPLE
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.
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);
CROSSREFS
KEYWORD
nonn
AUTHOR
Paolo P. Lava, Jul 08 2015
STATUS
approved