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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.
1
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
1,1
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.
LINKS
Javier Múgica, Table of n, a(n) for n = 1..19720 (terms <= 100000)
FORMULA
Asymptotic expression for a(n): e^(gamma)*n*(log(log(n))+O(1)), where gamma is Euler's constant: A001620.
EXAMPLE
The least prime factor of 77 is 7, and 77 < 2*3*5*7 = 210, therefore 77 belongs to the sequence.
PROG
(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
CROSSREFS
A279864 contains the same terms as this sequence in a different order.
Contains A308966. Both sequences agree in their first 38 terms.
Sequence in context: A192489 A161578 A261271 * A308966 A186891 A206074
KEYWORD
nonn
AUTHOR
Javier Múgica, May 30 2020
STATUS
approved