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

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, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229

Offset: 1

Author

Joshua R. Tint, Apr 04 2019

Keywords

• nonn
• easy

Comments

In other words, for every k > 1, there are at most 3 multiples of k in the sequence. - Rémy Sigrist, Apr 08 2019

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.

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.

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

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Aaron Greicius, The Prime Numbers, Lecture Fall 2012.

Crossrefs

See A166684 for the variant in which every divisor other than one is used at most twice.

Union of {1}, A000430 and A089581.

Programs

Maple

N:= 1000: # for terms <= N
M:= Vector(N):
Candidates:= {$2..N}:
A[1]:= 1:
for n from 2 while Candidates <> {} do
  A[n]:= min(Candidates):
  Candidates:= Candidates minus {A[n]};
  for d in numtheory:-divisors(A[n]) minus {1} do
     M[d]:= M[d]+1;
     if M[d] = 3 then Candidates:= Candidates minus {seq(i,i=2*d..N, d)} fi;
  od;
od:
seq(A[i],i=1..n-1); # Robert Israel, Apr 09 2019

Mathematica

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 *)

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

Extensions

More terms from Jinyuan Wang, Apr 07 2019