A370076 - OEIS

%I #15 Feb 11 2024 09:19:49

%S 4,8,9,12,18,20,24,25,27,28,32,40,44,45,49,50,52,54,56,60,63,68,75,76,

%T 84,88,90,92,96,98,99,104,116,117,120,121,124,125,126,128,132,135,136,

%U 140,147,148,150,152,153,156,160,164,168,169,171,172,175,184,188

%N Numbers k such that A005361(k) is prime.

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

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

%C 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... .

%H Amiram Eldar, <a href="/A370076/b370076.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

%Y Similar sequences: A009087, A023194.

%K nonn,easy

%O 1,1

%A _Amiram Eldar_, Feb 08 2024