A335284 Numbers k > 1 such that, if p is the least prime dividing k, k is less than or equal to the product of all prime numbers up to (and including) p.

2, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 119, 121, 127, 131, 133, 137, 139, 143, 149, 151, 157, 161, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 203

Offset: 1

Author

Javier Múgica, May 30 2020

Keywords

• nonn

Comments

The sequence A279864 contains the same terms as this one in different order, namely, sorted according to their least prime factor.

A number k > 1 belongs to this sequence if k <= A002110(A055396(k)) = A034386(A020639(k)). This condition approaches log(k) <= p as k -> infinity, p being the least prime factor of k.

All prime numbers belong to this sequence. Squares of prime numbers are included starting at 5^2; cubes are included starting at 11^3, and so on. That is, for all m there exists a p(m) such that all m-th powers of prime numbers from p(m)^m onwards belong to the sequence.

For large N the number of integers 1 < k <= N which belong to this sequence is ~ e^(-gamma)*N/log(log(N)), where gamma is Euler's constant: A001620.

Let p = p_r denote the r-th prime number and P_r = A034386(p) (the product of primes <= p). This sequence contains 1*2*4*...*(p_(r-1)-1) = A005867(r-1) elements whose least prime factor is p. These are distributed symmetrically about P_r/2, the first ones being p and, for p >= 5, p^2, and the last one being P_r-p.

Examples

			The least prime factor of 77 is 7, and 77 < 2*3*5*7 = 210, therefore 77 belongs to the sequence.
		

Links

• Javier Múgica, Table of n, a(n) for n = 1..19720 (terms <= 100000)

Crossrefs

Cf. A034386, A020639.

A279864 contains the same terms as this sequence in a different order.

Contains A308966. Both sequences agree in their first 38 terms.

Programs

PARI

isok(k) = if (k>1, my(p=vecmin(factor(k)[,1])); k <= prod(j=1, primepi(p), prime(j))); \\ Michel Marcus, May 31 2020

Formula

Asymptotic expression for a(n): e^(gamma)*n*(log(log(n))+O(1)), where gamma is Euler's constant: A001620.