A255733 - OEIS

%I #17 Mar 19 2015 08:42:10

%S 3,4,6,12,26,96,137,946,1053,2943,6874,17386,39182,60504,114254,

%T 167786,393216,497134,645354,5250086,27914146,448005874,505235234,

%U 708458286,3238952914,71258123714

%N Numbers x such that the sum of remainders of x mod k, where k runs through the anti-divisors of x, divides x.

%C For a(7) = 137, a(9) = 1053 and a(10) = 2943 the ratio is 1.

%C a(27) > 10^11. - _Hiroaki Yamanouchi_, Mar 17 2015

%e The anti-divisors of 26 are 3, 4, 17.

%e 26 mod 3 = 2; 26 mod 4 = 2; 26 mod 17 = 9; 2 + 2 + 9 = 13 and 26 / 13 = 2.

%p with(numtheory): P:=proc(q) local a,k,n;

%p for n from 3 to q do a:=0;

%p for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then a:=a+(n mod k);

%p fi; od; if type(n/a,integer) then print(n); fi; od; end: P(10^6);

%t f[n_] := Block[{ad}, ad[x_] := Cases[Range[2, x - 1], _?(Abs[Mod[x, #] - #/2] < 1 &)]; Plus @@ (Mod[n, #] & /@ ad@ n)]; Select[Range@ 5000, Mod[#, f@ #] == 0 &] (* _Michael De Vlieger_, Mar 05 2015 *)

%o (PARI) isok(n) = (n % sum(k=2, n-1, (n % k)*(abs((n % k)-k/2) < 1))) == 0; \\ _Michel Marcus_, Mar 06 2015

%Y Cf. A066272.

%K nonn,more

%O 1,1

%A _Paolo P. Lava_, Mar 05 2015

%E a(13)-a(26) from _Hiroaki Yamanouchi_, Mar 17 2015