A380733 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.

1, 2, 4, 8, 9, 16, 27, 32, 36, 64, 72, 81, 108, 125, 128, 144, 216, 243, 256, 288, 324, 432, 512, 576, 625, 648, 729, 864, 972, 1000, 1024, 1152, 1296, 1728, 1944, 2000, 2048, 2187, 2304, 2401, 2592, 2916, 3125, 3375, 3456, 3888, 4000, 4096, 4608, 5000, 5184, 5832

Offset: 1

Author

Amiram Eldar, Jan 31 2025

Keywords

• nonn

Comments

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

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

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).

Examples

			4 = 2^2 is a term since PrimePi(2) = 1 <= 2.
25 = 5^2 is not a term since PrimePi(5) = 3 is larger than the exponent 2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10226 (terms below 3*10^15)
• Eric Weisstein's World of Mathematics, Smooth Number.
• Wikipedia, Powerful number: Generalization (k-full number).
• Wikipedia, Smooth number.
• Index entries for sequences related to powerful numbers.

Crossrefs

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

Subsequences: A380730, A380731, A380732.

Programs

Mathematica

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

PARI

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

Formula

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.