A380733 - OEIS

%I #9 Jan 31 2025 13:39:38

%S 1,2,4,8,9,16,27,32,36,64,72,81,108,125,128,144,216,243,256,288,324,

%T 432,512,576,625,648,729,864,972,1000,1024,1152,1296,1728,1944,2000,

%U 2048,2187,2304,2401,2592,2916,3125,3375,3456,3888,4000,4096,4608,5000,5184,5832

%N Numbers k such that the prime index of the largest prime dividing k is smaller than or equal to the minimum exponent in the prime factorization of k; a(1) = 1 by convention.

%C Numbers k such that A061395(k) <= A051904(k).

%C Except for 2, all the terms are powerful (A001694).

%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 k-full but not (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 2, and S_k is infinite for k >= 2. The sum of the reciprocals of the terms of S_k is rational for all k: 1/2, 31/72, 2657/14400, ... (see the Formula section).

%H Amiram Eldar, <a href="/A380733/b380733.txt">Table of n, a(n) for n = 1..10226</a> (terms below 3*10^15)

%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) = 1.263759868201220952806..., 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/prime(j)^i), and S_k is defined in the Comments section.

%e 4 = 2^2 is a term since PrimePi(2) = 1 <= 2.

%e 25 = 5^2 is not a term since PrimePi(5) = 3 is larger than the exponent 2.

%t Select[Range[6000], Module[{f = FactorInteger[#]}, f[[-1, 1]] <= Prime[Min[f[[;;, 2]]]]] &]

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

%Y Cf. A001694, A006530, A051904, A061395, A380695.

%Y Subsequences: A380730, A380731, A380732.

%K nonn,new

%O 1,2

%A _Amiram Eldar_, Jan 31 2025