A380695 - OEIS

A380695

Numbers k such that the prime index of 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, 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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

Subsequence of A380692 and first differs from it at n = 428: A380692(428) = 625 is not a term of this sequence.

Numbers k such that A055396(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 = prime(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) = 0.68165919742420048618..., where d(i), the density of S_i, equals f(i+1) * Product_{primes p < prime(i)} ((1-1/p)/(1-1/p^(i+1))) - f(i) * Product_{primes p < prime(i)} ((1-1/p)/(1-1/p^i)), f(i) = 1/zeta(i) if i >= 2, and f(1) = 0.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

EXAMPLE

2 = 2^1 is a term since PrimePi(2) = 1 >= 1.

4 = 2^2 is not a term since PrimePi(2) = 1 is smaller than the exponent 2.

25 = 5^2 is a term since PrimePi(5) = 3 >= 2.

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A000720, A020639, A051903, A055396, A368110.