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

1,1

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... .

Links

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

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A020639, A051904.

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

Subsequence of A185359.

Sequence in context: A277780 A044893 A185359 * A022144 A181390 A008590

Adjacent sequences: A365883 A365884 A365885 * A365887 A365888 A365889

Keyword

nonn,easy

Author

Amiram Eldar, Sep 22 2023

Status

approved