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A054753 Numbers which are the product of a prime and the square of a different prime. 37
12, 18, 20, 28, 44, 45, 50, 52, 63, 68, 75, 76, 92, 98, 99, 116, 117, 124, 147, 148, 153, 164, 171, 172, 175, 188, 207, 212, 236, 242, 244, 245, 261, 268, 275, 279, 284, 292, 316, 325, 332, 333, 338, 356, 363, 369, 387, 388, 404, 412, 423, 425, 428, 436, 452 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A178254(a(n)) = 4; union of A095990 and A096156. - Reinhard Zumkeller, May 24 2010

Numbers with prime signature (2,1) = union of numbers with ordered prime signature (1,2) and numbers with ordered prime signature (2,1) (restating second part of above comment). - Daniel Forgues, Feb 05 2011

A056595(a(n)) = 4. - Reinhard Zumkeller, Aug 15 2011

Sum(n>=1, 1/a(n)^k) = P(k) * P(2*k) - P(3*k), where P is Prime Zeta function. - Enrique Pérez Herrero, Jun 27 2012

Also numbers n with A001222(n)=3 and A001221(n)=2. - Enrique Pérez Herrero, Jun 27 2012

A089233(a(n)) = 2. - Reinhard Zumkeller, Sep 04 2013

Numbers of the form p^2*q, with p,q distinct primes.  Subsequence of the triprimes (A014612).  If a(n) is even, then a(n)/2 is semiprime (A001358). - Wesley Ivan Hurt, Sep 08 2013

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Guilhem Castagnos, Antoine Joux, Fabien Laguillaumie, and Phong Q. Nguyen, Factoring pq^2 with quadratic forms: nice cryptanalyses, Advances in Cryptology - ASIACRYPT 2009. Lecture Notes in Computer Science Volume 5912 (2009), pp. 469-486.

René Peralta and Eiji Okamoto, Faster factoring of integers of a special form (1996)

StackExchange, Sequence of numbers with prime factorization pq^2

FORMULA

Solutions of the equation tau(n^5)=11*tau(n). - Paolo P. Lava, Mar 15 2013

EXAMPLE

a(1)=12 because 12=3*2*2.

MAPLE

with(numtheory);

A054753:=proc(q) local n;

for n from 1 to q do if tau(n^5)=11*tau(n) then print(n); fi; od; end:

A054753(10^10);  # Paolo P. Lava, Mar 15 2013

MATHEMATICA

Select[Range[12, 452], {1, 2}==Sort[Last/@FactorInteger[ # ]]&] (* Zak Seidov, Jul 19 2009 *)

With[{nn=60}, Take[Union[Flatten[{#[[1]]#[[2]]^2, #[[1]]^2 #[[2]]}&/@ Subsets[ Prime[Range[nn]], {2}]]], nn]] (* Harvey P. Dale, Dec 15 2014 *)

PROG

(PARI) is(n)=vecsort(factor(n)[, 2])==[1, 2]~ \\ Charles R Greathouse IV, Dec 30 2014

(PARI) for(n=1, 1e3, if(numdiv(n) - bigomega(n) == 3, print1(n, ", "))) \\ Altug Alkan, Nov 24 2015

CROSSREFS

Numbers with 6 divisors (A030515) which are not 5th powers of primes (A050997).

Sequence in context: A267117 A187039 A072357 * A098899 A098770 A181487

Adjacent sequences:  A054750 A054751 A054752 * A054754 A054755 A054756

KEYWORD

nonn

AUTHOR

Henry Bottomley, Apr 25 2000

EXTENSIONS

Link added and incorrect Mathematica code removed by David Bevan, Sep 17 2011

STATUS

approved

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Last modified December 6 15:11 EST 2016. Contains 278781 sequences.