A380694 - OEIS

A380694

Numbers k such that the prime index of the least prime dividing k is larger than the maximum exponent in the prime factorization of k.

3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 25, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 119, 121, 123, 127, 129, 131, 133, 137, 139, 141, 143, 145, 149

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OFFSET

1,1

COMMENTS

Differs from A352830 by having the terms 1225, 13475, 15925, ... and not having the terms 1, 147, 363, 507, 867, 1083, ... .

Differs from A320055 by having the terms 105, 165, 195, ... and not having the terms 1 and 2.

Differs from A320056 by having the terms 105, 165, 195, ... and not having the term 1.

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+1), 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) = 0.43156823896267860476......, 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

Eric Weisstein's World of Mathematics, Rough Number.

Wikipedia, Rough number.

EXAMPLE

3 = 3^1 is a term since PrimePi(3) = 2 > 1.

9 = 3^2 is not a term since PrimePi(3) = 2 is not larger 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[150], q]

PROG

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