login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 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 17-digit 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 11-digit 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 k-digit 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. 469-486.

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: A267117 A187039 A072357 * A098899 A098770 A181487

Adjacent sequences:  A054750 A054751 A054752 * A054754 A054755 A054756

KEYWORD

nonn,changed

AUTHOR

Henry Bottomley, Apr 25 2000

EXTENSIONS

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified September 24 07:50 EDT 2017. Contains 292403 sequences.