cp's OEIS Frontend

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.

%I A335284 #21 Jan 26 2021 03:39:10
%S A335284 2,3,5,7,11,13,17,19,23,25,29,31,37,41,43,47,49,53,59,61,67,71,73,77,
%T A335284 79,83,89,91,97,101,103,107,109,113,119,121,127,131,133,137,139,143,
%U A335284 149,151,157,161,163,167,169,173,179,181,187,191,193,197,199,203
%N 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.
%C A335284 The sequence A279864 contains the same terms as this one in different order, namely, sorted according to their least prime factor.
%C A335284 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.
%C A335284 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.
%C A335284 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.
%C A335284 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.
%H A335284 Javier Múgica, <a href="/A335284/b335284.txt">Table of n, a(n) for n = 1..19720</a> (terms <= 100000)
%F A335284 Asymptotic expression for a(n): e^(gamma)*n*(log(log(n))+O(1)), where gamma is Euler's constant: A001620.
%e A335284 The least prime factor of 77 is 7, and 77 < 2*3*5*7 = 210, therefore 77 belongs to the sequence.
%o A335284 (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
%Y A335284 Cf. A034386, A020639.
%Y A335284 A279864 contains the same terms as this sequence in a different order.
%Y A335284 Contains A308966. Both sequences agree in their first 38 terms.
%K A335284 nonn
%O A335284 1,1
%A A335284 _Javier Múgica_, May 30 2020