cp's OEIS Frontend

A343311 Numbers of the form x + y + z with distinct positive integers x,y,z such that (x+y+z) | x*y*z.

%I A343311 #11 Apr 13 2021 20:42:01
%S A343311 6,10,12,14,15,16,18,20,21,22,24,26,27,28,30,32,33,34,35,36,38,39,40,
%T A343311 42,44,45,46,48,49,50,51,52,54,55,56,57,58,60,62,63,64,65,66,68,69,70,
%U A343311 72,74,75,76,77,78,80,81,82,84,85,86,87,88,90,91,92,93,94,95,96,98,99,100
%N A343311 Numbers of the form x + y + z with distinct positive integers x,y,z such that (x+y+z) | x*y*z.
%C A343311 From _Robert Israel_, Apr 12 2021: (Start)
%C A343311 All terms are composite.
%C A343311 Conjecture: Consists of all composite numbers except 4, 8, 9, and 25. (End)
%H A343311 Robert Israel, <a href="/A343311/b343311.txt">Table of n, a(n) for n = 1..10000</a>
%e A343311 10 is in the sequence since 10 = 1+4+5 = 2+3+5, (1+4+5) | 1*4*5 and (2+3+5) | 2*3*5.
%e A343311 12 is in the sequence since 12 = 1+3+8 = 2+4+6 = 3+4+5, (1+3+8) | 1*3*8, (2+4+6) | 2*4*6 and (3+4+5) | 3*4*5.
%p A343311 filter:= proc(n) local x,y,z;
%p A343311   if isprime(n) then return false fi;
%p A343311   x:= min(numtheory:-factorset(n));
%p A343311   y:= n/x;
%p A343311   z:= n - x - y;
%p A343311   if z > 0 and nops({x,y,z}) = 3 then return true fi;
%p A343311   for x from 1 to n/3 do
%p A343311     for y from x+1 while x+2*y+1 <= n do
%p A343311       z:= n-x-y;
%p A343311       if x*y*z mod n = 0 then return true fi;
%p A343311   od od;
%p A343311   false
%p A343311 end proc:
%p A343311 select(filter, [$1..100]); # _Robert Israel_, Apr 12 2021
%t A343311 Table[If[Sum[Sum[(1 - KroneckerDelta[i, j]) (1 - KroneckerDelta[n - j, 2 i]) (1 - KroneckerDelta[n - i, 2 j]) (1 - Ceiling[i*j*(n - i - j)/n] + Floor[i*j*(n - i - j)/n]), {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}] > 0, n, {}], {n, 100}] // Flatten
%Y A343311 Cf. A343270.
%K A343311 nonn
%O A343311 1,1
%A A343311 _Wesley Ivan Hurt_, Apr 11 2021