A214842 Anti-multiply-perfect numbers. Numbers n for which sigma*(n)/n is an integer, where sigma*(n) is the sum of the anti-divisors of n.

1, 2, 5, 8, 41, 56, 77, 946, 1568, 2768, 5186, 6874, 8104, 17386, 27024, 84026, 167786, 2667584, 4775040, 4921776, 27914146, 505235234, 3238952914, 73600829714, 455879783074, 528080296234, 673223621664, 4054397778846, 4437083907194, 4869434608274, 6904301600914, 7738291969456

Offset: 1

Author

Paolo P. Lava, Mar 08 2013

Keywords

• nonn

Comments

A073930 and A073931 are subsets of this sequence.

Like A007691 but using sigma*(n) (A066417) instead of sigma(n) (A000203).

Tested up to 167786. Additional terms are 2667584, 4775040, 4921776, 27914146, 505235234, 3238952914, 73600829714 but there may be missing terms among them.

Examples

			Anti-divisors of 77 are 2, 3, 5, 9, 14, 17, 22, 31, 51. Their sum is 154 and 154/77=2.
		

Crossrefs

Cf. A000203, A007691, A066272, A066417, A073930, A073931.

Programs

Maple

A214842:= proc(q) local a,k,n;
for n from 1 to q do
  a:=0; for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then a:=a+k; fi; od;
  if type(a/n,integer) then print(n); fi; od; end:
A214842(10^10);

Mathematica

a066417[n_Integer] := Total[Cases[Range[2, n - 1], _?(Abs[Mod[n, #] - #/2] < 1 &)]]; a214842[n_Integer] := Select[Range[n], IntegerQ[a066417[#]/#] &];
a214842[1200] (* Michael De Vlieger, Aug 08 2014 *)

PARI

sad(n) = vecsum(select(t->n%t && tA066417
isok(n) = denominator(sad(n)/n) == 1; \\ Michel Marcus, Oct 12 2019

Python

A214842 = [n for n in range(1,10**4) if not (sum([d for d in range(2,n,2) if n%d and not 2*n%d])+sum([d for d in range(3,n,2) if n%d and 2*n%d in [d-1,1]])) % n]
# Chai Wah Wu, Aug 12 2014

Extensions

Verified there are no missing terms up to a(24) by Donovan Johnson, Apr 13 2013

a(25)-a(27) by Jud McCranie, Aug 31 2019

a(28)-a(32) by Jud McCranie, Oct 10 2019