A370076 Numbers k such that A005361(k) is prime.

4, 8, 9, 12, 18, 20, 24, 25, 27, 28, 32, 40, 44, 45, 49, 50, 52, 54, 56, 60, 63, 68, 75, 76, 84, 88, 90, 92, 96, 98, 99, 104, 116, 117, 120, 121, 124, 125, 126, 128, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 164, 168, 169, 171, 172, 175, 184, 188

Offset: 1

Author

Amiram Eldar, Feb 08 2024

Keywords

• nonn
• easy

Comments

Numbers of the form m * p^q, where p and q are primes, m is squarefree, and gcd(p, m) = 1.

The asymptotic density of this sequence is (6/Pi^2) * Sum_{p prime} ((p/(p+1)) * Sum_{q prime} 1/p^q) = 0.2933105687... .

The numbers k such that A005361(k) = 1 are the squarefree numbers (A005117), whose asymptotic density is 6/Pi^2 (A059956). The complement of the union of this sequence and the squarefree numbers is the sequence of numbers k such that A005361(k) is composite, whose asymptotic density is 0.0987623... .

Links

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

Crossrefs

Cf. A005117, A005361, A053810 (subsequence), A059956.

Similar sequences: A009087, A023194.

Programs

Mathematica

Select[Range[200], PrimeQ[Times @@ FactorInteger[#][[;;, 2]]] &]

PARI

is(n) = isprime(vecprod(factor(n)[ ,2]));