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A143610
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Numbers of the form p^2*q^3, where p,q are distinct primes.
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20
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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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
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FORMULA
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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
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EXAMPLE
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The first elements of this sequence are 3^2*2^3 = 72, 2^2*3^3 = 108, 5^2*2^3 = 200.
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MATHEMATICA
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f[n_] := Sort[Last/@FactorInteger[n]] == {2, 3}; Select[Range[30000], f] (* Vladimir Joseph Stephan Orlovsky, Oct 09 2009 *)
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PROG
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(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
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CROSSREFS
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Cf. A114128.
Cf. A085541, A085548, A085965.
Sequence in context: A072412 A052486 A114128 * A166987 A339940 A251124
Adjacent sequences: A143607 A143608 A143609 * A143611 A143612 A143613
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KEYWORD
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easy,nonn
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AUTHOR
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M. F. Hasler, Aug 27 2008
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STATUS
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approved
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