A189975 Numbers with prime factorization pqr^3 for distinct p, q, r.

120, 168, 264, 270, 280, 312, 378, 408, 440, 456, 520, 552, 594, 616, 680, 696, 702, 728, 744, 750, 760, 888, 918, 920, 945, 952, 984, 1026, 1032, 1064, 1128, 1144, 1160, 1240, 1242, 1272, 1288, 1416, 1464, 1480, 1485, 1496, 1566, 1608, 1624, 1640, 1672

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, May 03 2011

Keywords

• nonn

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. A030634, A050997, A178739.

Programs

Mathematica

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,3}; Select[Range[2000],f]

PARI

list(lim)=my(v=List(),t);forprime(p=2,(lim\6)^(1/3),forprime(q=2,sqrt(lim\p^3),if(p==q,next);t=p^3*q;forprime(r=q+1,lim\t,if(p==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 A189975(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:=x//r**3)))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)) for r in primerange(integer_nthroot(x,3)[0]+1))+sum(primepi(x//p**4) for p in primerange(integer_nthroot(x,4)[0]+1))-primepi(integer_nthroot(x,5)[0])
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025