A143610 Numbers of the form p^2 * q^3, where p,q are distinct primes.

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

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

Project Euler, Problem 200: Find the 200th prime-proof sqube containing the contiguous sub-string 200.

Index to sequences related to prime signature

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

Cf. A114128.

Cf. A085541, A085548, A085965.

Sequence in context: A375074 A375143 A375073 * A166987 A339940 A377854

Adjacent sequences: A143607 A143608 A143609 * A143611 A143612 A143613

Keyword

easy,nonn

Author

M. F. Hasler, Aug 27 2008

Status

approved