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

6, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100

Offset

1,1

Comments

From Robert Israel, Apr 12 2021: (Start)

All terms are composite.

Conjecture: Consists of all composite numbers except 4, 8, 9, and 25. (End)

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

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.
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.

Maple

filter:= proc(n) local x, y, z;
  if isprime(n) then return false fi;
  x:= min(numtheory:-factorset(n));
  y:= n/x;
  z:= n - x - y;
  if z > 0 and nops({x, y, z}) = 3 then return true fi;
  for x from 1 to n/3 do
    for y from x+1 while x+2*y+1 <= n do
      z:= n-x-y;
      if x*y*z mod n = 0 then return true fi;
  od od;
  false
end proc:
select(filter, [$1..100]); # Robert Israel, Apr 12 2021

Mathematica

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

Crossrefs

Cf. A343270.

Sequence in context: A361109 A317719 A329367 * A348718 A325265 A105642

Adjacent sequences: A343308 A343309 A343310 * A343312 A343313 A343314

Keyword

nonn

Author

Wesley Ivan Hurt, Apr 11 2021

Status

approved