A365886 Numbers k whose least prime divisor is smaller than its exponent in the prime factorization of k.

8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 81, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 243, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336, 344, 352, 360, 368, 376, 384, 392, 400, 405, 408, 416

Offset: 1

Author

Amiram Eldar, Sep 22 2023

Keywords

• nonn
• easy

Comments

First differs from A185359 at n = 22.

Numbers k such that A020639(k) < A051904(k).

The asymptotic density of terms with least prime factor prime(n) (within all the positive integers) is d(n) = (1/prime(n)^(prime(n)+1)) * Product_{k=1..(n-1)} (1-1/prime(k)). For example, for n = 1, 2, 3, 4 and 5, d(n) = 1/8, 1/162, 1/46875, 4/86472015 and 8/109844993185235.

The asymptotic density of this sequence is Sum_{n>=1} d(n) = 0.13119421909731920416... .

Examples

			8 = 2^3 is a term since its least prime factor, 2, is smaller than its exponent, 3.
		

Links

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

Crossrefs

Cf. A020639, A051904.

Subsequences: A008590 \ {0}, A365887, A365888.

Subsequence of A185359.

Programs

Mathematica

q[n_] := Less @@ FactorInteger[n][[1]]; Select[Range[2, 420], q]

PARI

is(n) = {my(f = factor(n)); n > 1 && f[1, 1] < f[1, 2];}