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

1,1

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.

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A190108, A190109.

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

Sequence in context: A391448 A223457 A338541 * A049205 A234426 A187436

Adjacent sequences: A190374 A190375 A190376 * A190378 A190379 A190380

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, May 09 2011

Status

approved