A380730 - OEIS

%I #11 Apr 26 2025 05:31:20

%S 8,16,32,64,81,128,243,256,512,729,1024,1296,2048,2187,2592,3888,4096,

%T 5184,6561,7776,8192,10368,11664,15552,15625,16384,19683,20736,23328,

%U 31104,32768,34992,41472,46656,59049,62208,65536,69984,78125,82944,93312,104976,124416

%N Numbers k such that the greatest prime dividing k is smaller than the minimum exponent in the prime factorization of k.

%C Numbers k such that A006530(k) < A051904(k).

%C 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)+1)-full but not (prime(k+1)+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 8, and S_k is infinite for k >= 2. The sum of the reciprocals of the terms of S_k is rational for all k: 1/8, 583/5184, 19757609/777600000, ... (see the Formula section).

%H Amiram Eldar, <a href="/A380730/b380730.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SmoothNumber.html">Smooth Number</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Powerful_number#Generalization">Powerful number: Generalization</a> (k-full number).

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Smooth_number">Smooth number</a>.

%H <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.

%F Sum_{n>=1} 1/a(n) = Sum_{k>=1} f(k) = 0.27091620709274155136..., 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} 1/prime(j)^i), and S_k is defined in the Comments section.

%e 8 = 2^3 is a term since 2 < 3.

%e 9 = 3^2 is not a term since 3 > 2.

%t Select[Range[2, 125000], Module[{f = FactorInteger[#]}, f[[-1, 1]] < Min[f[[;;, 2]]]] &]

%o (PARI) isok(k) = if(k == 1, 0, my(f = factor(k), e = f[,2]); f[#f~, 1] < vecmin(e));

%Y Cf. A006530, A051904, A380692.

%Y Subsequence of A036966, A380731, A380732 and A380733.

%K nonn

%O 1,1

%A _Amiram Eldar_, Jan 31 2025