

A054753


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


45



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
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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 the 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
From Bernard Schott, Sep 16 2017: (Start)
These numbers are called "Nombres d'Einstein" on the French site "Diophante" (see link) because a(n) = m * c^2 where m and c are two different primes.
The numbers 44 = 2^2 * 11 and 45 = 3^2 * 5 are the two smallest consecutive "Einstein numbers"; 603, 604, 605 are the three smallest consecutive integers in this sequence. It's not possible to get more than five such consecutive numbers (proof in the link); the first set of five such consecutive numbers begins at the 17digit number 10093613546512321. Where does the first sequence of four consecutive "Einstein numbers" begin? (End) [corrected by Jon E. Schoenfield, Sep 20 2017]
The first set of four consecutive integers in this sequence begins at the 11digit number 17042641441. (Each such set must include two even numbers, one of which is of the form 2^2*q, the other of the form p^2*2; a quick search, taking the factorizations of consecutive integers before and after numbers of the latter form, shows that the number of sets of four consecutive kdigit integers in this sequence is 1, 7, 12, 18 for k = 11, 12, 13, 14, respectively.)  Jon E. Schoenfield, Sep 16 2017
The first 13 sets of 5 consecutive integers in this sequence have as their first terms 10093613546512321, 14414905793929921, 266667848769941521, 562672865058083521, 1579571757660876721, 1841337567664174321, 2737837351207392721, 4456162869973433521, 4683238426747860721, 4993613853242910721, 5037980611623036721, 5174116847290255921, 5344962129269790721. Each of these numbers except the last is 7^2 times a prime; the last is 23^2 times a prime.  Jon E. Schoenfield, Sep 17 2017


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. 469486.
Michel Lafond, Les Nombres d'Einstein, Diophante, A 350, May 2014
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 = 2^2*3 is the smallest number of the form p^2*q.


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

Cf. A001221, A001222, A001358, A014612, A056595, A089233, A095990, A096156, A178254.
Numbers with 6 divisors (A030515) which are not 5th powers of primes (A050997).
Sequence in context: A187039 A325241 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



