A380730 Numbers k such that the greatest prime dividing k is smaller than the minimum exponent in the prime factorization of k.

8, 16, 32, 64, 81, 128, 243, 256, 512, 729, 1024, 1296, 2048, 2187, 2592, 3888, 4096, 5184, 6561, 7776, 8192, 10368, 11664, 15552, 15625, 16384, 19683, 20736, 23328, 31104, 32768, 34992, 41472, 46656, 59049, 62208, 65536, 69984, 78125, 82944, 93312, 104976, 124416

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)+1)-full but not (prime(k+1)+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 8, and S_k is infinite for k >= 2. The sum of the reciprocals of the terms of S_k is rational for all k: 1/8, 583/5184, 19757609/777600000, ... (see the Formula section).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

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) = 0.27091620709274155136..., 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} 1/prime(j)^i), and S_k is defined in the Comments section.

Example

8 = 2^3 is a term since 2 < 3.
9 = 3^2 is not a term since 3 > 2.

Mathematica

Select[Range[2, 125000], 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, A380692.

Subsequence of A036966, A380731, A380732 and A380733.

Sequence in context: A371011 A277128 A054743 * A192135 A345053 A256817

Adjacent sequences: A380727 A380728 A380729 * A380731 A380732 A380733

Keyword

nonn

Author

Amiram Eldar, Jan 31 2025

Status

approved