A179643 Products of exactly 2 distinct squares of primes and a different prime (p^2 * q^2 * r).

180, 252, 300, 396, 450, 468, 588, 612, 684, 700, 828, 882, 980, 1044, 1100, 1116, 1300, 1332, 1452, 1476, 1548, 1575, 1692, 1700, 1900, 1908, 2028, 2124, 2156, 2178, 2196, 2205, 2300, 2412, 2420, 2450, 2475, 2548, 2556, 2628, 2844, 2900, 2925, 2988

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

Keywords

• nonn

Comments

A050326(a(n)) = 5, subsequence of A225228. - Reinhard Zumkeller, May 03 2013

Examples

			180 = 2^2 * 3^2 * 5, 252 = 2^2 * 3^2 * 7, 300 = 2^2 * 3 * 5^2, ...
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Will Nicholes, List of Prime Signatures
• Index to sequences related to prime signature

Crossrefs

Cf. A006881, A007304, A065036, A085986, A085987, A178739, A179642.

Cf. A179695.

Programs

Mathematica

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,2,2}; Select[Range[3000], f]

PARI

list(lim)=my(v=List(),t);forprime(p=2,sqrt(lim\12),forprime(q=p+1,sqrt(lim\p^2\2),t=(p*q)^2;forprime(r=2,lim\t,if(p==r||q==r,next);listput(v,t*r))));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 19 2011

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A179643(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((t:=primepi(s:=isqrt(y:=isqrt(x//r))))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)) for r in primerange(x+1))+sum(primepi(isqrt(x//p**3)) for p in primerange(integer_nthroot(x,3)[0]+1))-primepi(integer_nthroot(x,5)[0])
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025