A214842 - OEIS

%I #38 Dec 07 2019 12:18:26

%S 1,2,5,8,41,56,77,946,1568,2768,5186,6874,8104,17386,27024,84026,

%T 167786,2667584,4775040,4921776,27914146,505235234,3238952914,

%U 73600829714,455879783074,528080296234,673223621664,4054397778846,4437083907194,4869434608274,6904301600914,7738291969456

%N 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 A073930 and A073931 are subsets of this sequence.

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

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

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

%p A214842:= proc(q) local a,k,n;

%p for n from 1 to q do

%p   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   if type(a/n,integer) then print(n); fi; od; end:

%p A214842(10^10);

%t a066417[n_Integer] := Total[Cases[Range[2, n - 1], _?(Abs[Mod[n, #] - #/2] < 1 &)]]; a214842[n_Integer] := Select[Range[n], IntegerQ[a066417[#]/#] &];

%t a214842[1200] (* _Michael De Vlieger_, Aug 08 2014 *)

%o (Python)

%o 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 # _Chai Wah Wu_, Aug 12 2014

%o (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 isok(n) = denominator(sad(n)/n) == 1; \\ _Michel Marcus_, Oct 12 2019

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

%K nonn

%O 1,2

%A _Paolo P. Lava_, Mar 08 2013

%E Verified there are no missing terms up to a(24) by _Donovan Johnson_, Apr 13 2013

%E a(25)-a(27) by _Jud McCranie_, Aug 31 2019

%E a(28)-a(32) by _Jud McCranie_, Oct 10 2019