A380692 Numbers k such that the least prime dividing k is larger than the maximum exponent in the prime factorization of k; a(1) = 1 by convention.

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 99

Offset: 1

Author

Amiram Eldar, Jan 30 2025

Keywords

• nonn
• easy

Comments

First differs from A368110 at n = 68: A368110(68) = 98 is not a term of this sequence.

First differs from its subsequence A380695 at n = 428: a(428) = 625 is not a term of A380695.

Numbers k such that A020639(k) > A051903(k).

All the squarefree numbers (A005117) are terms, and all the odd terms of A067259 are terms of this sequence.

Disjoint union of the sequences S_k, k >= 1, where S_k is the sequence of p-rough numbers (numbers whose prime factors are all greater than or equal to p), with p = nextprime(k) = A151800(k), whose maximum exponent in their prime factorization is k (i.e., numbers that are (k+1)-free but not k-free, where k-free numbers are numbers whose prime factorization exponents do not exceed k).

The asymptotic density of this sequence is Sum_{i>=1} d(i) = 0.68213349032332767778..., where d(i), the density of S_i, equals f(i+1) * Product_{primes p <= i} ((1-1/p)/(1-1/p^(i+1))) - f(i) * Product_{primes p <= i} ((1-1/p)/(1-1/p^i)), f(i) = 1/zeta(i) if i >= 2, and f(1) = 0.

Examples

			6 = 2^1 * 3^1 is a term since 2 > 1.
8 = 2^3 is not a term since 2 < 3.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Rough Number.
• Wikipedia, Rough number.

Crossrefs

Cf. A020639, A051903, A151800, A368110.

Subsequence of A380693.

Subsequences: A005117, A136327, the intersection of A005408 and A067259, A380694, A380695.

Programs

Mathematica

q[k_] := k == 1 || Module[{f = FactorInteger[k]}, f[[1, 1]] > Max[f[[;; , 2]]]]; Select[Range[100], q]

PARI

isok(k) = if(k == 1, 0, my(f = factor(k), e = f[, 2]); f[1, 1] > vecmax(e));