A291046 Minimal multiplicative semigroup of numbers n > 1 such that in the prime factorization of n an initial product of primes is greater than a later prime in the factorization.

30, 60, 70, 90, 105, 120, 140, 150, 154, 165, 180, 182, 195, 210, 231, 240, 270, 273, 280, 286, 300, 308, 315, 330, 350, 357, 360, 364, 374, 385, 390, 399, 418, 420, 429, 442, 450, 455, 462, 480, 490, 494, 495, 510, 525, 540, 546, 560, 561, 570, 572, 585, 595, 598, 600, 616, 627

Offset: 1

Author

Richard Locke Peterson, Aug 16 2017

Keywords

• nonn

Comments

Definition: Let a number n>1 have prime factorization n=p1^e1*...*pi^ei*..*pm^em, with the primes written in ascending order and the ei>0. If an initial product p1*..*pi is greater than some later prime p(i+1), then n is in the sequence. The definition contains a more restrictive requirement than A289484 does, so it is a proper subsemigroup of A289484. It can be seen that if s and t are in the sequence, the so is s*t. More strongly, if n is in the sequence, so is every multiple of n. Any number in it is divisible by at least 3 primes, although that is not a sufficient condition.

Differs from A212666 first at a(93), because 930=2*3*5*31 is in this sequence but not in A212666. - R. J. Mathar, Sep 02 2018

Numbers whose squarefree kernel (A007947) is in A164596. - Peter Munn, Feb 05 2024

Links

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

Crossrefs

Cf. A007947, A164596, A212666, A289484.

Programs

Maple

filter:= proc(n) local S,p,i;
  S:= sort(convert(numtheory:-factorset(n),list));
  p:= 1;
  for i from 1 to nops(S)-1 do
    p:= p*S[i];
    if p > S[i+1] then return true fi;
  od;
  false
end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 26 2018