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.

4, 8, 16, 27, 32, 64, 81, 128, 216, 243, 256, 432, 512, 625, 648, 729, 864, 1024, 1296, 1728, 1944, 2048, 2187, 2592, 3125, 3456, 3888, 4096, 5184, 5832, 6561, 6912, 7776, 8192, 10000, 10368, 11664, 13824, 15552, 15625, 16384, 16807, 17496, 19683, 20000, 20736

Offset: 1

Author

Amiram Eldar, Jan 31 2025

Keywords

• nonn

Comments

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

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).

Examples

			4 = 2^2 is a term since PrimePi(2) = 1 < 2.
9 = 3^2 is not a term since PrimePi(3) = 2 is not larger than the exponent 2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..12735 (terms below 10^18)
• 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. A006530, A051904, A061395, A380694.

Subsequence of A001694 and A380733.

Subsequences: A380730, A380731.

Programs

Mathematica

Select[Range[2, 33000], Module[{f = FactorInteger[#]}, f[[-1, 1]] < Prime[Min[f[[;;, 2]]]]] &]

PARI

isok(k) = if(k == 1, 0, 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) = 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.