A365889 Numbers k whose least prime divisor divides its exponent in the prime factorization of k.

4, 12, 16, 20, 27, 28, 36, 44, 48, 52, 60, 64, 68, 76, 80, 84, 92, 100, 108, 112, 116, 124, 132, 135, 140, 144, 148, 156, 164, 172, 176, 180, 188, 189, 192, 196, 204, 208, 212, 220, 228, 236, 240, 244, 252, 256, 260, 268, 272, 276, 284, 292, 297, 300, 304, 308

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) = ((prime(n)-1)/(prime(n)*(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) = 1/6, 1/78, 1/11715, 4/14411985 and 8/10984499318485.

The asymptotic density of this sequence is Sum_{n>=1} d(n) = 0.17957281768342725732... .

Examples

			4 = 2^2 is a term since its least prime factor, 2, divides its exponent, 2.
16 = 2^4 is a term since its least prime factor, 2, divides its exponent, 4.
		

Links

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

Crossrefs

Cf. A020639, A051904.

Subsequence of A342090.

Subsequences: A365883, A365890, A365891.

Programs

Mathematica

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

PARI

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