A179698 Numbers of the form p^4*q^3*r where p, q, and r are distinct primes.

2160, 3024, 3240, 4536, 4752, 5616, 6000, 7128, 7344, 8208, 8424, 9936, 11016, 12312, 12528, 13392, 14000, 14904, 15000, 15984, 16464, 17712, 18576, 18792, 20088, 20250, 20304, 22000, 22896, 23976, 25488, 26000, 26352, 26568, 27440

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Jul 24 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

Programs

Mathematica

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,3,4}; Select[Range[30000], f]

PARI

list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\24)^(1/4), t1=p^4;forprime(q=2, (lim\t1)^(1/3), if(p==q, next);t2=t1*q^3;forprime(r=2, lim\t2, if(p==r||q==r, next);listput(v,t2*r)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011

Python

from sympy import primepi, primerange, integer_nthroot
def A179698(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**3)) for p in primerange(integer_nthroot(x,4)[0]+1) for q in primerange(integer_nthroot(x//p**4,3)[0]+1))+sum(primepi(integer_nthroot(x//p**4,4)[0]) for p in primerange(integer_nthroot(x,4)[0]+1))+sum(primepi(integer_nthroot(x//p**5,3)[0]) for p in primerange(integer_nthroot(x,5)[0]+1))+sum(primepi(x//p**7) for p in primerange(integer_nthroot(x,7)[0]+1))-(primepi(integer_nthroot(x,8)[0])<<1)
    return bisection(f,n,n) # Chai Wah Wu, Mar 28 2025