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

1,1

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).

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.

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.

Example

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.

Mathematica

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

Prog

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

Crossrefs

Cf. A006530, A051904, A061395, A380694.

Subsequence of A001694 and A380733.

Subsequences: A380730, A380731.

Sequence in context: A325127 A054744 A100391 * A380731 A122494 A257278

Adjacent sequences: A380729 A380730 A380731 * A380733 A380734 A380735

Keyword

nonn

Author

Amiram Eldar, Jan 31 2025

Status

approved