A190377 Numbers with prime factorization p^2*q^2*r^2*s^2 where p, q, r, and s are distinct primes.

44100, 108900, 152100, 213444, 260100, 298116, 324900, 476100, 509796, 592900, 636804, 736164, 756900, 828100, 864900, 933156, 1232100, 1258884, 1334025, 1416100, 1483524, 1512900, 1572516, 1664100, 1695204, 1758276, 1768900, 1863225

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, May 09 2011

Keywords

• nonn

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Will Nicholes, List of prime signatures, 2010.
• Index to sequences related to prime signature.

Crossrefs

Cf. A190108, A190109.

Cf. A046386, A085548, A085964, A085966, A085968.

Programs

Mathematica

f[n_]:=Sort[Last/@FactorInteger[n]]=={2,2,2,2};Select[Range[3000000],f]

PARI

list(lim)=my(v=List(),t1,t2,t3); forprime(p=2,sqrtint(lim\900), t1=p^2; forprime(q=2,sqrtint(lim\(36*t1)), if(q==p, next); t2=q^2*t1; forprime(r=2,sqrtint(lim\(4*t2)), if(r==p || r==q, next); t3=r^2*t2; forprime(s=2,sqrtint(lim\t3), if(s==p || s==q || s==r, next); listput(v, t3*s^2))))); Set(v) \\ Charles R Greathouse IV, Aug 25 2016

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A190377(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 int(n+x-sum(primepi(x//(k*m*r))-c for a,k in enumerate(primerange(integer_nthroot(x,4)[0]+1),1) for b,m in enumerate(primerange(k+1,integer_nthroot(x//k,3)[0]+1),a+1) for c,r in enumerate(primerange(m+1,isqrt(x//(k*m))+1),b+1)))
    return bisection(f,n,n)**2 # Chai Wah Wu, Mar 27 2025

Formula

Sum_{n>=1} 1/a(n) = (P(2)^4 - 6*P(2)^2*P(4) + 8*P(2)*P(6) + 3*P(4)^2 - 6*P(8))/24 = 0.00010511750849230980748..., where P is the prime zeta function. - Amiram Eldar, Mar 07 2024

a(n) = A046386(n)^2. - Chai Wah Wu, Mar 27 2025