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A380732 Numbers k such that the prime index of the largest prime dividing k is smaller than the minimum exponent in the prime factorization of k.

%I A380732 #7 Jan 31 2025 13:39:32
%S A380732 4,8,16,27,32,64,81,128,216,243,256,432,512,625,648,729,864,1024,1296,
%T A380732 1728,1944,2048,2187,2592,3125,3456,3888,4096,5184,5832,6561,6912,
%U A380732 7776,8192,10000,10368,11664,13824,15552,15625,16384,16807,17496,19683,20000,20736
%N A380732 Numbers k such that the prime index of the largest prime dividing k is smaller than the minimum exponent in the prime factorization of k.
%C A380732 Numbers k such that A061395(k) < A051904(k).
%C A380732 Disjoint union of the sequences S_k, k >= 1, where S_k is the sequence of p-smooth numbers (numbers whose prime factors are all less than or equal to p), with p = prime(k), that are (k+1)-full but not (k+2)-full numbers (k-full numbers are numbers whose prime factorization exponents are all larger than or equal to k). S_1 contains only the term 4, and S_k is infinite for k >= 2. The sum of the reciprocals of the terms of S_k is rational for all k: 1/4, 25/144, 509579/6480000, ... (see the Formula section).
%H A380732 Amiram Eldar, <a href="/A380732/b380732.txt">Table of n, a(n) for n = 1..12735</a> (terms below 10^18)
%H A380732 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SmoothNumber.html">Smooth Number</a>.
%H A380732 Wikipedia, <a href="https://en.wikipedia.org/wiki/Powerful_number#Generalization">Powerful number: Generalization</a> (k-full number).
%H A380732 Wikipedia, <a href="https://en.wikipedia.org/wiki/Smooth_number">Smooth number</a>.
%H A380732 <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.
%F A380732 Sum_{n>=1} 1/a(n) = Sum_{k>=1} f(k) = 0.57181100946173735203..., where f(k) = Sum_{i>=1} 1 / S_k(i) = g(k, k) - g(k+1, k), g(e, k) = Product_{j=1..k} (1 + Sum_{i >= e+1} 1/prime(j)^i), and S_k is defined in the Comments section.
%e A380732 4 = 2^2 is a term since PrimePi(2) = 1 < 2.
%e A380732 9 = 3^2 is not a term since PrimePi(3) = 2 is not larger than the exponent 2.
%t A380732 Select[Range[2, 33000], Module[{f = FactorInteger[#]}, f[[-1, 1]] < Prime[Min[f[[;;, 2]]]]] &]
%o A380732 (PARI) isok(k) = if(k == 1, 0, my(f = factor(k), e = f[,2]); f[#f~, 1] < prime(vecmin(e)));
%Y A380732 Cf. A006530, A051904, A061395, A380694.
%Y A380732 Subsequence of A001694 and A380733.
%Y A380732 Subsequences: A380730, A380731.
%K A380732 nonn
%O A380732 1,1
%A A380732 _Amiram Eldar_, Jan 31 2025