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A054753 Numbers which are the product of a prime and the square of a different prime (p^2*q). 40
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

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

Index to sequences related to prime signature

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 July 25 21:59 EDT 2017. Contains 289798 sequences.