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%I #70 Nov 30 2023 01:24:39
%S 72,108,200,392,500,675,968,1125,1323,1352,1372,2312,2888,3087,3267,
%T 4232,4563,5324,6125,6728,7688,7803,8575,8788,9747,10952,11979,13448,
%U 14283,14792,15125,17672,19652,19773,21125,22472,22707,25947,27436
%N Numbers of the form p^2 * q^3, where p,q are distinct primes.
%C Also: numbers with prime signature {3,2}.
%C This is a subsequence of A114128. [Hasler]
%C Every a(n) is an Achilles number (A052486). They are minimal, meaning no proper divisor is an Achilles number. - _Antonio Roldán_, Dec 27 2011
%H T. D. Noe, <a href="/A143610/b143610.txt">Table of n, a(n) for n = 1..1000</a>
%H Project Euler, <a href="https://projecteuler.net/problem=200">Problem 200: Find the 200th prime-proof sqube containing the contiguous sub-string 200</a>.
%H <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%F 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
%e The first three terms of this sequence are 3^2 * 2^3 = 72, 2^2 * 3^3 = 108, 5^2 * 2^3 = 200.
%t f[n_] := Sort[Last/@FactorInteger[n]] == {2, 3}; Select[Range[30000], f] (* _Vladimir Joseph Stephan Orlovsky_, Oct 09 2009 *)
%o (PARI) for(n=1, 10^5, omega(n)==2 || next; vecsort(factor(n)[,2])==[2,3]~ && print1(n","))
%o (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
%Y Cf. A114128.
%Y Cf. A085541, A085548, A085965.
%K easy,nonn
%O 1,1
%A _M. F. Hasler_, Aug 27 2008
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