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

1,1

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

Mathematica

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

Prog

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

Crossrefs

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

Similar sequences: A009087, A023194.

Sequence in context: A119025 A167903 A074661 * A252849 A082293 A048944

Adjacent sequences: A370073 A370074 A370075 * A370077 A370078 A370079

Keyword

nonn,easy

Author

Amiram Eldar, Feb 08 2024

Status

approved