A179644 Product of the 4th power of a prime and 2 different distinct primes (p^4*q*r).

240, 336, 528, 560, 624, 810, 816, 880, 912, 1040, 1104, 1134, 1232, 1360, 1392, 1456, 1488, 1520, 1776, 1782, 1840, 1904, 1968, 2064, 2106, 2128, 2256, 2288, 2320, 2480, 2544, 2576, 2754, 2832, 2835, 2928, 2960, 2992, 3078, 3216, 3248, 3280, 3344, 3408

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

Keywords

• nonn

Examples

			240=2^4*3*5,336=2^4*3*7,..810=2^3^4*5,..
		

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, A179643.

Programs

Mathematica

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,4}; Select[Range[4000], f]
Take[Union[#[[1]]^4 #[[2]]#[[3]]&/@(Flatten[Permutations/@ Subsets[ Prime[ Range[ 20]],{3}],1])],50] (* Harvey P. Dale, Feb 07 2013 *)

PARI

list(lim)=my(v=List(),t);forprime(p=2,(lim\6)^(1/4),forprime(q=2,sqrt(lim\p^4),if(p==q,next);t=p^4*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 A179644(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**4)))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)) for r in primerange(integer_nthroot(x,4)[0]+1))+sum(primepi(x//p**5) for p in primerange(integer_nthroot(x,5)[0]+1))-primepi(integer_nthroot(x,6)[0])
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025