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A096156 Numbers with ordered prime signature (2,1). 13
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
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
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

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Last modified March 28 16:58 EDT 2024. Contains 371254 sequences. (Running on oeis4.)