A229861 Let sigma*_m (n) be result of applying sum of anti-divisors m times to n; call n (m,k)-anti-perfect if sigma*_m (n) = k*n; sequence gives the (3,k)-anti-perfect numbers.
4, 5, 8, 32, 41, 54, 56, 68, 123, 946, 1494, 1856, 2056, 5186, 6874, 8104, 10419, 17386, 27024, 31100, 84026, 167786, 272089, 733253, 812600, 1188000, 1544579, 2667584, 4921776, 16360708, 21524990, 27914146
Offset: 1
Examples
Anti-divisors of 54 are 4, 12, 36. Their sum is 52. Again, anti-divisors of 52 are 3, 5, 7, 8, 15, 21, 35. Their sum is 94. Finally, anti-divisors of 94 are 3, 4, 7, 9, 11, 17, 21, 27, 63. Their sum is 162 and 162 / 54 = 3.
Programs
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Maple
with(numtheory); P:=proc(q,h) local a,i,j,k,n; for n from 4 to q do a:=n; for i from 1 to h do k:=0; j:=a; while j mod 2 <> 1 do k:=k+1; j:=j/2; od; a:=sigma(2*a+1)+sigma(2*a-1)+sigma(a/2^k)*2^(k+1)-6*a-2; od; if type(a/n,integer) then print(n); fi; od; end: P(10^6,3);
Extensions
Offset corrected and a(26)-a(32) from Donovan Johnson, Jan 09 2014
Comments