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A380731 Numbers k such that the largest prime dividing k is smaller than or equal to the minimum exponent in the prime factorization of k.

%I A380731 #9 Jan 31 2025 13:39:25
%S A380731 4,8,16,27,32,64,81,128,216,243,256,432,512,648,729,864,1024,1296,
%T A380731 1728,1944,2048,2187,2592,3125,3456,3888,4096,5184,5832,6561,6912,
%U A380731 7776,8192,10368,11664,13824,15552,15625,16384,17496,19683,20736,23328,27648,31104,32768
%N A380731 Numbers k such that the largest prime dividing k is smaller than or equal to the minimum exponent in the prime factorization of k.
%C A380731 Numbers k such that A006530(k) <= A051904(k).
%C A380731 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 prime(k)-full but not prime(k+1)-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, 649/2592, 61992313/1166400000, ... (see the Formula section).
%H A380731 Amiram Eldar, <a href="/A380731/b380731.txt">Table of n, a(n) for n = 1..5416</a> (terms below 10^18)
%H A380731 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SmoothNumber.html">Smooth Number</a>.
%H A380731 Wikipedia, <a href="https://en.wikipedia.org/wiki/Powerful_number#Generalization">Powerful number: Generalization</a> (k-full number).
%H A380731 Wikipedia, <a href="https://en.wikipedia.org/wiki/Smooth_number">Smooth number</a>.
%H A380731 <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.
%F A380731 Sum_{n>=1} 1/a(n) = Sum_{k>=1} f(k) = 0.56987350769329353172..., where f(k) = Sum_{i>=1} 1 / S_k(i) = g(prime(k), k) - g(prime(k+1), k), g(p, k) = Product_{j=1..k} (1 + Sum_{i >= p} 1/prime(j)^i), and S_k is defined in the Comments section.
%e A380731 4 = 2^2 is a term since A006530(4) = A051904(4) = 2.
%e A380731 9 = 3^2 is not a term since 3 > 2.
%p A380731 filter:= proc(n) local F;
%p A380731   F:= ifactors(n)[2];
%p A380731   max(F[..,1]) <= min(F[..,2])
%p A380731 end proc:
%p A380731 select(filter, [$2..50000]); # _Robert Israel_, Jan 31 2025
%t A380731 Select[Range[2, 33000], Module[{f = FactorInteger[#]}, f[[-1, 1]] <= Min[f[[;;, 2]]]] &]
%o A380731 (PARI) isok(k) = if(k == 1, 0, my(f = factor(k), e = f[,2]); f[#f~, 1] <= vecmin(e));
%Y A380731 Cf. A006530, A051904, A380693.
%Y A380731 Subsequence of A001694, A380732 and A380733.
%Y A380731 A380730 is a subsequence.
%K A380731 nonn
%O A380731 1,1
%A A380731 _Amiram Eldar_, Jan 31 2025