cp's OEIS Frontend

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.

%I A380695 #9 Jan 31 2025 04:24:30
%S A380695 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 A380695 37,38,39,41,42,43,45,46,47,49,51,53,55,57,58,59,61,62,63,65,66,67,69,
%U A380695 70,71,73,74,75,77,78,79,82,83,85,86,87,89,91,93,94,95,97,99
%N 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.
%C A380695 First differs from A368110 at n = 68: A368110(68) = 98 is not a term of this sequence.
%C A380695 Subsequence of A380692 and first differs from it at n = 428: A380692(428) = 625 is not a term of this sequence.
%C A380695 Numbers k such that A055396(k) >= A051903(k).
%C A380695 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).
%C A380695 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.
%H A380695 Amiram Eldar, <a href="/A380695/b380695.txt">Table of n, a(n) for n = 1..10000</a>
%e A380695 2 = 2^1 is a term since PrimePi(2) = 1 >= 1.
%e A380695 4 = 2^2 is not a term since PrimePi(2) = 1 is smaller than the exponent 2.
%e A380695 25 = 5^2 is a term since PrimePi(5) = 3 >= 2.
%t A380695 q[k_] := k == 1 || Module[{f = FactorInteger[k]}, f[[1, 1]] >= Prime[Max[f[[;; , 2]]]]]; Select[Range[100], q]
%o A380695 (PARI) isok(k) = if(k == 1, 1, my(f = factor(k), e = f[,2]); f[1,1] >= prime(vecmax(e)));
%Y A380695 Cf. A000720, A020639, A051903, A055396, A368110.
%Y A380695 Subsequence of A380692 and A380693.
%Y A380695 A380694 is a subsequence.
%K A380695 nonn,easy
%O A380695 1,2
%A A380695 _Amiram Eldar_, Jan 30 2025