A365883 Numbers k whose least prime divisor is equal to its exponent in the prime factorization of k.

4, 12, 20, 27, 28, 36, 44, 52, 60, 68, 76, 84, 92, 100, 108, 116, 124, 132, 135, 140, 148, 156, 164, 172, 180, 188, 189, 196, 204, 212, 220, 228, 236, 244, 252, 260, 268, 276, 284, 292, 297, 300, 308, 316, 324, 332, 340, 348, 351, 356, 364, 372, 380, 388, 396

Offset: 1

Author

Amiram Eldar, Sep 22 2023

Keywords

• nonn
• easy

Comments

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/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, 1/81, 4/46875, 8/28824005 and 16/21968998637047.

The asymptotic density of this sequence is Sum_{n>=1} d(n) = 0.13743128989284883653... .

Examples

			4 = 2^2 is a term since its least prime factor, 2, is equal to its exponent.
		

Links

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

Crossrefs

Cf. A020639, A051904.

Subsequence of A100717 and A365889.

Subsequences: A017113, A365884, A365885.

Programs

Maple

filter:= proc(n) local F;
   F:= sort(ifactors(n)[2],(s,t) -> s[1]Robert Israel, Sep 22 2023

Mathematica

q[n_] := Equal @@ FactorInteger[n][[1]]; Select[Range[2, 400], q]

PARI

is(n) = n > 1 && #Set(factor(n)[1,]) == 1;