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

1,1

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A030634, A050997, A178739.

Sequence in context: A396323 A272594 A377156 * A232461 A090782 A337386

Adjacent sequences: A189972 A189973 A189974 * A189976 A189977 A189978

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, May 03 2011

Status

approved