A380731 Numbers k such that the largest prime dividing k is smaller than or equal to the minimum exponent in the prime factorization of k.

4, 8, 16, 27, 32, 64, 81, 128, 216, 243, 256, 432, 512, 648, 729, 864, 1024, 1296, 1728, 1944, 2048, 2187, 2592, 3125, 3456, 3888, 4096, 5184, 5832, 6561, 6912, 7776, 8192, 10368, 11664, 13824, 15552, 15625, 16384, 17496, 19683, 20736, 23328, 27648, 31104, 32768

Offset

1,1

Comments

Numbers k such that A006530(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 prime(k)-full but not prime(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 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, 649/2592, 61992313/1166400000, ... (see the Formula section).

Links

Amiram Eldar, Table of n, a(n) for n = 1..5416 (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) = 0.56987350769329353172..., 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/prime(j)^i), and S_k is defined in the Comments section.

Example

4 = 2^2 is a term since A006530(4) = A051904(4) = 2.
9 = 3^2 is not a term since 3 > 2.

Maple

filter:= proc(n) local F;
  F:= ifactors(n)[2];
  max(F[.., 1]) <= min(F[.., 2])
end proc:
select(filter, [$2..50000]); # Robert Israel, Jan 31 2025

Mathematica

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

Prog

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

Crossrefs

Cf. A006530, A051904, A380693.

Subsequence of A001694, A380732 and A380733.

A380730 is a subsequence.

Sequence in context: A054744 A100391 A380732 * A122494 A257278 A257279

Adjacent sequences: A380728 A380729 A380730 * A380732 A380733 A380734

Keyword

nonn

Author

Amiram Eldar, Jan 31 2025

Status

approved