|
%I #29 Dec 20 2021 10:01:13
%S 12,20,28,44,45,52,63,68,76,92,99,116,117,124,148,153,164,171,172,175,
%T 188,207,212,236,244,261,268,275,279,284,292,316,325,332,333,356,369,
%U 387,388,404,412,423,425,428,436,452,475,477,508,524,531,539,548,549
%N Numbers with ordered prime signature (2,1).
%C Numbers of the form p^2 * q where p and q are primes with p < q.
%C Also terms of A054753 that are not in A095990.
%C 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.
%H Enrique Pérez Herrero, <a href="/A096156/b096156.txt">Table of n, a(n) for n = 1..5000</a>
%H OEIS Wiki, <a href="http://oeis.org/wiki/Ordered_prime_signatures">Ordered prime signatures</a>
%e a(2) = 20 because 20 = 2*2*5 and 2 < 5.
%e 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).
%t Take[ Sort[ Flatten[ Table[ Prime[p]^2 Prime[q], {q, 2, 33}, {p, q - 1}]]], 54] (* _Robert G. Wilson v_, Jul 28 2004 *)
%t Select[Range[10^5],FactorInteger[#][[All,2]]=={2,1}&] (* _Enrique Pérez Herrero_, Jun 27 2012 *)
%o (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
%o (Python)
%o from sympy import factorint
%o def ok(n): return list(factorint(n).values()) == [2, 1]
%o print([k for k in range(550) if ok(k)]) # _Michael S. Branicky_, Dec 20 2021
%Y Cf. A095990.
%Y Subsequence of A054753, A097320, A325241, A345381.
%K nonn,easy
%O 1,1
%A _Alford Arnold_, Jul 24 2004
%E Edited and extended by _Robert G. Wilson v_ and _Rick L. Shepherd_, Jul 27 2004
|
