A096156 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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.

LINKS

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000

OEIS Wiki, Ordered prime signatures

EXAMPLE

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).

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 *)

PROG

(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

CROSSREFS

Cf. A095990.

Subsequence of A054753, A097320, A325241, A345381.

Sequence in context: A360767 A366825 A110187 * A364999 A366807 A210968

Adjacent sequences: A096153 A096154 A096155 * A096157 A096158 A096159

KEYWORD

nonn,easy

AUTHOR

Alford Arnold, Jul 24 2004

EXTENSIONS

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

STATUS

approved