A380693 Numbers k such that the least prime dividing k is larger than or equal to the maximum exponent in the prime factorization of k; a(1) = 1 by convention.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Author

Amiram Eldar, Jan 30 2025

Keywords

• nonn
• easy

Comments

First differs from A047592, A187320, A207481 and A255805 at n = 48: A047592(48) = A187320(48) = A207481(48) = A255805(48) = 54 is not a term of this sequence.

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

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) = A007918(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) = 0.84999238500582943243..., 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, A007918.

Cf. A047592, A187320, A207481, A255805.

Subsequences: A380692, 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));