[go: up one dir, main page]

login
A143610
Numbers of the form p^2 * q^3, where p,q are distinct primes.
24
72, 108, 200, 392, 500, 675, 968, 1125, 1323, 1352, 1372, 2312, 2888, 3087, 3267, 4232, 4563, 5324, 6125, 6728, 7688, 7803, 8575, 8788, 9747, 10952, 11979, 13448, 14283, 14792, 15125, 17672, 19652, 19773, 21125, 22472, 22707, 25947, 27436
OFFSET
1,1
COMMENTS
Also: numbers with prime signature {3,2}.
This is a subsequence of A114128. [Hasler]
Every a(n) is an Achilles number (A052486). They are minimal, meaning no proper divisor is an Achilles number. - Antonio Roldán, Dec 27 2011
FORMULA
Sum_{n>=1} 1/a(n) = P(2)*P(3) - P(5) = A085548 * A085541 - A085965 = 0.043280..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020
EXAMPLE
The first three terms of this sequence are 3^2 * 2^3 = 72, 2^2 * 3^3 = 108, 5^2 * 2^3 = 200.
MATHEMATICA
f[n_] := Sort[Last/@FactorInteger[n]] == {2, 3}; Select[Range[30000], f] (* Vladimir Joseph Stephan Orlovsky, Oct 09 2009 *)
PROG
(PARI) for(n=1, 10^5, omega(n)==2 || next; vecsort(factor(n)[, 2])==[2, 3]~ && print1(n", "))
(PARI) list(lim)=my(v=List(), t); forprime(p=2, (lim\4)^(1/3), t=p^3; forprime(q=2, sqrt(lim\t), if(p==q, next); listput(v, t*q^2))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
M. F. Hasler, Aug 27 2008
STATUS
approved