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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.

%I A380692 #10 Jan 31 2025 04:24:41
%S A380692 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,
%T A380692 37,38,39,41,42,43,45,46,47,49,51,53,55,57,58,59,61,62,63,65,66,67,69,
%U A380692 70,71,73,74,75,77,78,79,82,83,85,86,87,89,91,93,94,95,97,99
%N 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.
%C A380692 First differs from A368110 at n = 68: A368110(68) = 98 is not a term of this sequence.
%C A380692 First differs from its subsequence A380695 at n = 428: a(428) = 625 is not a term of A380695.
%C A380692 Numbers k such that A020639(k) > A051903(k).
%C A380692 All the squarefree numbers (A005117) are terms, and all the odd terms of A067259 are terms of this sequence.
%C A380692 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).
%C A380692 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.
%H A380692 Amiram Eldar, <a href="/A380692/b380692.txt">Table of n, a(n) for n = 1..10000</a>
%H A380692 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RoughNumber.html">Rough Number</a>.
%H A380692 Wikipedia, <a href="https://en.wikipedia.org/wiki/Rough_number">Rough number</a>.
%e A380692 6 = 2^1 * 3^1 is a term since 2 > 1.
%e A380692 8 = 2^3 is not a term since 2 < 3.
%t A380692 q[k_] := k == 1 || Module[{f = FactorInteger[k]}, f[[1, 1]] > Max[f[[;; , 2]]]]; Select[Range[100], q]
%o A380692 (PARI) isok(k) = if(k == 1, 0, my(f = factor(k), e = f[, 2]); f[1, 1] > vecmax(e));
%Y A380692 Cf. A020639, A051903, A151800, A368110.
%Y A380692 Subsequence of A380693.
%Y A380692 Subsequences: A005117, A136327, the intersection of A005408 and A067259, A380694, A380695.
%K A380692 nonn,easy
%O A380692 1,2
%A A380692 _Amiram Eldar_, Jan 30 2025