A179669 Products of form p^4*q^2*r where p, q and r are three distinct primes.

720, 1008, 1200, 1584, 1620, 1872, 2268, 2352, 2448, 2736, 2800, 3312, 3564, 3920, 4050, 4176, 4212, 4400, 4464, 5200, 5328, 5508, 5808, 5904, 6156, 6192, 6768, 6800, 7452, 7500, 7600, 7632, 7938, 8112, 8496, 8624, 8784, 9200, 9396, 9648, 9680, 10044

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Jul 23 2010

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. A137493.

Programs

Maple

N:= 20000: # for terms < N
P:= select(isprime,[2,seq(i,i=1..N/(3^2*2^4),2)]):
R:= NULL:
for i from 1 while P[i]^4 * 2^2*3 < N do
  for j from 1 while P[i]^4 * P[j]^2 *2 < N do
    if j = i then next fi;
    m:= ListTools:-BinaryPlace(P,N/P[i]^4/P[j]^2);
    R:= R, seq(P[i]^4*P[j]^2*P[k],k={$1..m} minus {i,j});
od od:
sort([R]); # Robert Israel, Mar 28 2025

Mathematica

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,2,4}; Select[Range[10000], f]

PARI

list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\12)^(1/4), t1=p^4;forprime(q=2, sqrt(lim\t1), if(p==q, next);t2=t1*q^2;forprime(r=2, lim\t2, if(p==r||q==r, next);listput(v,t2*r)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A179669(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**4*q**2)) for p in primerange(integer_nthroot(x,4)[0]+1) for q in primerange(isqrt(x//p**4)+1))+sum(primepi(integer_nthroot(x//p**4,3)[0]) for p in primerange(integer_nthroot(x,4)[0]+1))+sum(primepi(isqrt(x//p**5)) for p in primerange(integer_nthroot(x,5)[0]+1))+sum(primepi(x//p**6) for p in primerange(integer_nthroot(x,6)[0]+1))-(primepi(integer_nthroot(x,7)[0])<<1)
    return bisection(f,n,n) # Chai Wah Wu, Mar 28 2025