A342090 Numbers with at least one prime power p^e in their prime factorization such that p|e.

4, 12, 16, 20, 27, 28, 36, 44, 48, 52, 54, 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, 216, 220, 228, 236, 240, 244, 252, 256, 260, 268, 270, 272, 276, 284, 292, 297, 300

Offset: 1

Author

Amiram Eldar, Feb 27 2021

Keywords

• nonn

Comments

Numbers with a unitary divisor of the form p^(m*p) where p is a prime and m > 0.

The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 1, 19, 188, 1883, 18825, 188244, 1882429, 18824297, 188242957, 1882429628, ...

The asymptotic density of this sequence is 1 - Product_{p prime} 1 - (p - 1)/(p*(p^p - 1)) = 0.18824296270011399086...

Examples

			4 = 2^2 is a term since 2 divides 2.
8 = 2^3 is not a term since 2 does not divide 3.
		

Links

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

Crossrefs

Subsequence of A013929.

Subsequences: A000302, A000312, A009971, A017113, A051674.

Cf. A072873, A369070 (characteristic function).

Programs

Mathematica

q[n_] := AnyTrue[FactorInteger[n], Divisible[Last[#], First[#]] &]; Select[Range[2, 300], q]

Extensions

Wrong term 1 removed by Amiram Eldar, Jan 16 2024