OFFSET
1,1
COMMENTS
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
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
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 for 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.
Diophante, A 350, Les Nombres d'Einstein (in French).
Mathematics Stack Exchange, Sequence of numbers with prime factorization pq^2.
René Peralta and Eiji Okamoto, Faster factoring of integers of a special form (1996).
FORMULA
A056595(a(n)) = 4. - Reinhard Zumkeller, Aug 15 2011
A089233(a(n)) = 2. - Reinhard Zumkeller, Sep 04 2013
a(n) ~ k*n log n where k = 1/A085548 = 2.211179.... - Charles R Greathouse IV, Feb 04 2026
EXAMPLE
a(1) = 12 because 12 = 2^2*3 is the smallest number of the form p^2*q.
MATHEMATICA
Select[Range[12, 452], {1, 2}==Sort[Last/@FactorInteger[ # ]]&] (* Zak Seidov, Jul 19 2009 *)
(* Alternative: *)
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
(PARI) list(lim) = { my(v=List()); forprime(q=2, lim, forprime(p=2, sqrtint(lim\q), if(p!=q, listput(v, p^2*q)))); Set(v) }; \\ Bruce Nye, May 30 2026
(Python)
from sympy import factorint
def ok(n): return sorted(factorint(n).values()) == [1, 2]
print([k for k in range(453) if ok(k)]) # Michael S. Branicky, Dec 18 2021
(Python)
from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A054753(n):
def bisection(f, kmin=0, kmax=1):
while f(kmax) > kmax: kmax <<= 1
kmin = kmax >> 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
def f(x): return n+x-sum(primepi(x//p**2) for p in primerange(isqrt(x)+1))+primepi(integer_nthroot(x, 3)[0])
return bisection(f, n, n) # Chai Wah Wu, Feb 21 2025
CROSSREFS
Table giving for each subsequence the corresponding number of groups of order p^2*q, from Bernard Schott, Jan 23 2022
-------------------------------------------------------------------------------
-------------------------------------------------------------------------------
|A000001(p^2*q)| (q+9)/2 | 5 | 5 | 4 | 3 | 2 |
-------------------------------------------------------------------------------
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