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

1,2

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.

Links

Table of n, a(n) for n=1..32.

Example

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

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 *)

Prog

(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
(PARI) sad(n) = vecsum(select(t->n%t && t<n, concat(concat(divisors(2*n-1), divisors(2*n+1)), 2*divisors(n)))); \\ A066417
isok(n) = denominator(sad(n)/n) == 1; \\ Michel Marcus, Oct 12 2019

Crossrefs

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

Sequence in context: A323461 A191550 A120342 * A299517 A268660 A180627

Adjacent sequences: A214839 A214840 A214841 * A214843 A214844 A214845

Keyword

nonn

Author

Paolo P. Lava, Mar 08 2013

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

Status

approved