A096156 Numbers with ordered prime signature (2,1).

12, 20, 28, 44, 45, 52, 63, 68, 76, 92, 99, 116, 117, 124, 148, 153, 164, 171, 172, 175, 188, 207, 212, 236, 244, 261, 268, 275, 279, 284, 292, 316, 325, 332, 333, 356, 369, 387, 388, 404, 412, 423, 425, 428, 436, 452, 475, 477, 508, 524, 531, 539, 548, 549

Offset: 1

Author

Alford Arnold, Jul 24 2004

Keywords

• nonn
• easy

Comments

Numbers of the form p^2 * q where p and q are primes with p < q.

Also terms of A054753 that are not in A095990.

There are pairs that differ by 1, which is not the case in A095990, beginning with 44 and 45, 116 and 117, 171 and 172, 332 and 333, etc.

Examples

			a(2) = 20 because 20 = 2*2*5 and 2 < 5.
Note that 18 = 2*3^2 is not in the sequence, even though it has prime signature (2,1), because its ordered prime signature is (1,2) (A095990). Prime signatures correspond to partitions of Omega(n), while ordered prime signatures correspond to compositions of Omega(n).
		

Links

• Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
• OEIS Wiki, Ordered prime signatures

Crossrefs

Cf. A095990.

Subsequence of A054753, A097320, A325241, A345381.

Programs

Mathematica

Take[ Sort[ Flatten[ Table[ Prime[p]^2 Prime[q], {q, 2, 33}, {p, q - 1}]]], 54] (* Robert G. Wilson v, Jul 28 2004 *)
Select[Range[10^5],FactorInteger[#][[All,2]]=={2,1}&] (* Enrique Pérez Herrero, Jun 27 2012 *)

PARI

list(lim)=my(v=List()); forprime(q=3, lim\4, forprime(p=2, min(sqrtint(lim\q), q-1), listput(v, p^2*q))); Set(v) \\ Charles R Greathouse IV, Feb 26 2014

Python

from sympy import factorint
def ok(n): return list(factorint(n).values()) == [2, 1]
print([k for k in range(550) if ok(k)]) # Michael S. Branicky, Dec 20 2021

Extensions

Edited and extended by Robert G. Wilson v and Rick L. Shepherd, Jul 27 2004