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

1,1

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

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

Crossrefs

Cf. A007947, A164596, A212666, A289484.

Sequence in context: A051283 A066031 A212666 * A071140 A221054 A365795

Adjacent sequences: A291043 A291044 A291045 * A291047 A291048 A291049

Keyword

nonn

Author

Richard Locke Peterson, Aug 16 2017

Status

approved