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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.

%I A214842 #38 Dec 07 2019 12:18:26
%S A214842 1,2,5,8,41,56,77,946,1568,2768,5186,6874,8104,17386,27024,84026,
%T A214842 167786,2667584,4775040,4921776,27914146,505235234,3238952914,
%U A214842 73600829714,455879783074,528080296234,673223621664,4054397778846,4437083907194,4869434608274,6904301600914,7738291969456
%N 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.
%C A214842 A073930 and A073931 are subsets of this sequence.
%C A214842 Like A007691 but using sigma*(n) (A066417) instead of sigma(n) (A000203).
%C A214842 Tested up to 167786. Additional terms are 2667584, 4775040, 4921776, 27914146, 505235234, 3238952914, 73600829714 but there may be missing terms among them.
%e A214842 Anti-divisors of 77 are 2, 3, 5, 9, 14, 17, 22, 31, 51. Their sum is 154 and 154/77=2.
%p A214842 A214842:= proc(q) local a,k,n;
%p A214842 for n from 1 to q do
%p A214842   a:=0; for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then a:=a+k; fi; od;
%p A214842   if type(a/n,integer) then print(n); fi; od; end:
%p A214842 A214842(10^10);
%t A214842 a066417[n_Integer] := Total[Cases[Range[2, n - 1], _?(Abs[Mod[n, #] - #/2] < 1 &)]]; a214842[n_Integer] := Select[Range[n], IntegerQ[a066417[#]/#] &];
%t A214842 a214842[1200] (* _Michael De Vlieger_, Aug 08 2014 *)
%o A214842 (Python)
%o A214842 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]
%o A214842 # _Chai Wah Wu_, Aug 12 2014
%o A214842 (PARI) sad(n) = vecsum(select(t->n%t && t<n, concat(concat(divisors(2*n-1), divisors(2*n+1)), 2*divisors(n)))); \\ A066417
%o A214842 isok(n) = denominator(sad(n)/n) == 1; \\ _Michel Marcus_, Oct 12 2019
%Y A214842 Cf. A000203, A007691, A066272, A066417, A073930, A073931.
%K A214842 nonn
%O A214842 1,2
%A A214842 _Paolo P. Lava_, Mar 08 2013
%E A214842 Verified there are no missing terms up to a(24) by _Donovan Johnson_, Apr 13 2013
%E A214842 a(25)-a(27) by _Jud McCranie_, Aug 31 2019
%E A214842 a(28)-a(32) by _Jud McCranie_, Oct 10 2019