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A307360 A sequence in which every divisor other than 1 is used at most three times.

%I A307360 #53 Apr 29 2021 01:43:12
%S A307360 1,2,3,4,5,6,7,9,11,13,17,19,23,25,29,31,35,37,41,43,47,49,53,59,61,
%T A307360 67,71,73,79,83,89,97,101,103,107,109,113,121,127,131,137,139,143,149,
%U A307360 151,157,163,167,169,173,179,181,191,193,197,199,211,223,227,229
%N A307360 A sequence in which every divisor other than 1 is used at most three times.
%C A307360 In other words, for every k > 1, there are at most 3 multiples of k in the sequence. - _Rémy Sigrist_,  Apr 08 2019
%C A307360 The sequence begins at 1. The smallest integer greater than the last term which is not divisible by a divisor already used three times (excluding one) is added to the sequence.
%C A307360 Contains all prime numbers (A000040), given that the prime numbers only have the divisors of themselves and one, by definition, therefore the only divisor which could exist in the sequence already to disqualify the number from inclusion in the sequence would be the prime number itself, but a number cannot have a divisor higher than itself (the prime numbers), so given that the sequence increases, the divisor could not exist in the sequence, and any prime number would be included.
%C A307360 Terms are {1} or primes or squares of primes (A000430) or numbers of the form prime(2k + 1) * prime(2k + 2) (A089581) where k >= 0. - _David A. Corneth_, Apr 09 2019
%H A307360 Robert Israel, <a href="/A307360/b307360.txt">Table of n, a(n) for n = 1..10000</a>
%H A307360 Aaron Greicius, <a href="http://gauss.math.luc.edu/greicius/Math201/Fall2012/Lectures/primes.article.pdf">The Prime Numbers</a>, Lecture Fall 2012.
%e A307360 For instance, 8 is not in the sequence because 2, 4, and 6 are all divisible by 2 and appear previously in the sequence. The sequence, then, skips to nine. After 9, no more numbers divisible by three appear in the sequence, given that after 3 and 6, it is the third number divisible by three to appear in the sequence.
%p A307360 N:= 1000: # for terms <= N
%p A307360 M:= Vector(N):
%p A307360 Candidates:= {$2..N}:
%p A307360 A[1]:= 1:
%p A307360 for n from 2 while Candidates <> {} do
%p A307360   A[n]:= min(Candidates):
%p A307360   Candidates:= Candidates minus {A[n]};
%p A307360   for d in numtheory:-divisors(A[n]) minus {1} do
%p A307360      M[d]:= M[d]+1;
%p A307360      if M[d] = 3 then Candidates:= Candidates minus {seq(i,i=2*d..N, d)} fi;
%p A307360   od;
%p A307360 od:
%p A307360 seq(A[i],i=1..n-1); # _Robert Israel_, Apr 09 2019
%t A307360 Select[Range@ 229, Or[# == 1, PrimeQ@ #, PrimeQ@ Sqrt@ #, And[SquareFreeQ@ #, If[PrimeNu@ # == 2, And[OddQ@ First@ #, Apply[SameQ, (# - {1, 2})/2]] &@ PrimePi[FactorInteger[#][[All, 1]]], False]]] &] (* _Michael De Vlieger_, Apr 11 2019 *)
%o A307360 (PARI) is(n) = if(n==1, return(1)); my(f=factor(n)); if(f[, 2] == [1]~ || f[, 2] ==[2]~, return(1)); if(f[,2] == [1,1]~ && nextprime(f[1,1]+1) == f[2,1] && primepi(f[1,1]) % 2 == 1, return(1)); 0 \\ _David A. Corneth_, Apr 09 2019
%Y A307360 See A166684 for the variant in which every divisor other than one is used at most twice.
%Y A307360 Union of {1}, A000430 and A089581.
%K A307360 nonn,easy
%O A307360 1,2
%A A307360 _Joshua R. Tint_, Apr 04 2019
%E A307360 More terms from _Jinyuan Wang_, Apr 07 2019