cp's OEIS Frontend

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

%I A054753 #131 Feb 21 2025 13:07:45
%S A054753 12,18,20,28,44,45,50,52,63,68,75,76,92,98,99,116,117,124,147,148,153,
%T A054753 164,171,172,175,188,207,212,236,242,244,245,261,268,275,279,284,292,
%U A054753 316,325,332,333,338,356,363,369,387,388,404,412,423,425,428,436,452
%N A054753 Numbers which are the product of a prime and the square of a different prime (p^2 * q).
%C A054753 A178254(a(n)) = 4; union of A095990 and A096156. - _Reinhard Zumkeller_, May 24 2010
%C A054753 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
%C A054753 A056595(a(n)) = 4. - _Reinhard Zumkeller_, Aug 15 2011
%C A054753 For k>1, 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
%C A054753 Also numbers n with A001222(n)=3 and A001221(n)=2. - _Enrique Pérez Herrero_, Jun 27 2012
%C A054753 A089233(a(n)) = 2. - _Reinhard Zumkeller_, Sep 04 2013
%C A054753 Subsequence of the triprimes (A014612). If a(n) is even, then a(n)/2 is semiprime (A001358). - _Wesley Ivan Hurt_, Sep 08 2013
%C A054753 From _Bernard Schott_, Sep 16 2017: (Start)
%C A054753 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.
%C A054753 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]
%C A054753 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
%C A054753 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 for the last is 7^2 times a prime; the last is 23^2 times a prime. - _Jon E. Schoenfield_, Sep 17 2017
%H A054753 Reinhard Zumkeller, <a href="/A054753/b054753.txt">Table of n, a(n) for n = 1..1000</a>
%H A054753 Guilhem Castagnos, Antoine Joux, Fabien Laguillaumie, and Phong Q. Nguyen, <a href="http://www.iacr.org/archive/asiacrypt2009/59120468/59120468.pdf">Factoring pq^2 with quadratic forms: nice cryptanalyses</a>, Advances in Cryptology - ASIACRYPT 2009. Lecture Notes in Computer Science Volume 5912 (2009), pp. 469-486.
%H A054753 Diophante, <a href="http://www.diophante.fr/problemes-par-themes/arithmetique-et-algebre/a3-nombres-remarquables/2854-a350-les-nombres-deinstein">A 350, Les Nombres d'Einstein</a> (in French).
%H A054753 Mathematics Stack Exchange, <a href="http://math.stackexchange.com/questions/64892/sequence-of-numbers-with-prime-factorization-pq2">Sequence of numbers with prime factorization pq^2</a>
%H A054753 René Peralta and Eiji Okamoto, <a href="https://citeseerx.ist.psu.edu/pdf/d091fc5d18d572ff9a602ac526f2f3dabf84851e">Faster factoring of integers of a special form</a> (1996).
%H A054753 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%e A054753 a(1) = 12 because 12 = 2^2*3 is the smallest number of the form p^2*q.
%t A054753 Select[Range[12,452], {1,2}==Sort[Last/@FactorInteger[ # ]]&] (* Zak Seidov, Jul 19 2009 *)
%t A054753 With[{nn=60},Take[Union[Flatten[{#[[1]]#[[2]]^2,#[[1]]^2 #[[2]]}&/@ Subsets[ Prime[Range[nn]],{2}]]],nn]] (* _Harvey P. Dale_, Dec 15 2014 *)
%o A054753 (PARI) is(n)=vecsort(factor(n)[,2])==[1,2]~ \\ _Charles R Greathouse IV_, Dec 30 2014
%o A054753 (PARI) for(n=1, 1e3, if(numdiv(n) - bigomega(n) == 3, print1(n, ", "))) \\ _Altug Alkan_, Nov 24 2015
%o A054753 (Python)
%o A054753 from sympy import factorint
%o A054753 def ok(n): return sorted(factorint(n).values()) == [1, 2]
%o A054753 print([k for k in range(453) if ok(k)]) # _Michael S. Branicky_, Dec 18 2021
%o A054753 (Python)
%o A054753 from math import isqrt
%o A054753 from sympy import primepi, primerange, integer_nthroot
%o A054753 def A054753(n):
%o A054753     def bisection(f,kmin=0,kmax=1):
%o A054753         while f(kmax) > kmax: kmax <<= 1
%o A054753         kmin = kmax >> 1
%o A054753         while kmax-kmin > 1:
%o A054753             kmid = kmax+kmin>>1
%o A054753             if f(kmid) <= kmid:
%o A054753                 kmax = kmid
%o A054753             else:
%o A054753                 kmin = kmid
%o A054753         return kmax
%o A054753     def f(x): return n+x-sum(primepi(x//p**2) for p in primerange(isqrt(x)+1))+primepi(integer_nthroot(x,3)[0])
%o A054753     return bisection(f,n,n) # _Chai Wah Wu_, Feb 21 2025
%Y A054753 Cf. A001221, A001222, A001358, A014612, A056595, A089233, A095990, A096156, A178254.
%Y A054753 Numbers with 6 divisors (A030515) which are not 5th powers of primes (A050997).
%Y A054753 Subsequence of A325241.  Supersequence of A096156.
%Y A054753 Table giving for each subsequence the corresponding number of groups of order p^2*q, from _Bernard Schott_, Jan 23 2022
%Y A054753   -------------------------------------------------------------------------------
%Y A054753   | Subsequence  | A350638 | A143928 | A350115 | A349495 | A350245 | A350422 (*)|
%Y A054753   -------------------------------------------------------------------------------
%Y A054753   |A000001(p^2*q)| (q+9)/2 |    5    |    5    |    4    |    3    |      2     |
%Y A054753   -------------------------------------------------------------------------------
%Y A054753 (*) A350422 equals disjoint union of A350332 (p<q) and A350421 (p>q).
%K A054753 nonn
%O A054753 1,1
%A A054753 _Henry Bottomley_, Apr 25 2000
%E A054753 Link added and incorrect Mathematica code removed by _David Bevan_, Sep 17 2011