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

Author

Amiram Eldar, Jan 31 2025

Keywords

• nonn

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).

Examples

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

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.

Crossrefs

Cf. A006530, A051904, A380693.

Subsequence of A001694, A380732 and A380733.

A380730 is a subsequence.

Programs

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]]]] &]

PARI

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

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.