cp's OEIS Frontend

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.

%I A291046 #24 Feb 05 2024 18:10:35
%S A291046 30,60,70,90,105,120,140,150,154,165,180,182,195,210,231,240,270,273,
%T A291046 280,286,300,308,315,330,350,357,360,364,374,385,390,399,418,420,429,
%U A291046 442,450,455,462,480,490,494,495,510,525,540,546,560,561,570,572,585,595,598,600,616,627
%N 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.
%C A291046 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.
%C A291046 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
%C A291046 Numbers whose squarefree kernel (A007947) is in A164596. - _Peter Munn_, Feb 05 2024
%H A291046 Robert Israel, <a href="/A291046/b291046.txt">Table of n, a(n) for n = 1..10000</a>
%p A291046 filter:= proc(n) local S,p,i;
%p A291046   S:= sort(convert(numtheory:-factorset(n),list));
%p A291046   p:= 1;
%p A291046   for i from 1 to nops(S)-1 do
%p A291046     p:= p*S[i];
%p A291046     if p > S[i+1] then return true fi;
%p A291046   od;
%p A291046   false
%p A291046 end proc:
%p A291046 select(filter, [$1..1000]); # _Robert Israel_, Aug 26 2018
%Y A291046 Cf. A007947, A164596, A212666, A289484.
%K A291046 nonn
%O A291046 1,1
%A A291046 _Richard Locke Peterson_, Aug 16 2017