A385288 Numbers with a prime number of prime factors, counted with multiplicity, and whose prime factors are each raised to a prime exponent.

4, 8, 9, 25, 27, 32, 49, 72, 108, 121, 125, 128, 169, 200, 243, 288, 289, 343, 361, 392, 500, 529, 675, 800, 841, 961, 968, 972, 1125, 1323, 1331, 1352, 1369, 1372, 1568, 1681, 1800, 1849, 2048, 2187, 2197, 2209, 2312, 2700, 2809, 2888, 3087, 3125, 3267, 3481

Offset: 1

Author

James C. McMahon, Jun 24 2025

Keywords

• nonn

Comments

a(n) = A114129(n) through n=25; then a(26) = 961 and A114129(26) = 864.

Subset of A056166.

Subset of A001694. - Michael De Vlieger, Jun 25 2025.

Examples

			200 = 2^3 * 5^2; 200 has a prime number of prime factors, counted with multiplicity (3 + 2 = 5), and exponents 3 and 2 are prime.
		

Links

• James C. McMahon, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A001248, A001694, A030078, A056166, A114129.

Programs

Mathematica

Select[Range[10^4],AllTrue[Last/@FactorInteger[#],PrimeQ]&&PrimeQ[PrimeOmega[#]]&]

PARI

isok(k) = my(f=factor(k)); isprime(bigomega(k)) && (sum(k=1, #f~, isprime(f[k,2])) == omega(f)); \\ Michel Marcus, Jun 25 2025