A190115 Numbers with prime factorization p^2*q^3*r^4 where p, q, and r are distinct primes.
10800, 16200, 18000, 21168, 31752, 40500, 45000, 49392, 52272, 67500, 73008, 78408, 98000, 109512, 111132, 124848, 137200, 155952, 172872, 187272, 191664, 228528, 233928, 242000, 245000, 259308, 316368, 338000, 342792, 363312, 415152
Offset: 1
Keywords
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
Programs
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Mathematica
f[n_]:=Sort[Last/@FactorInteger[n]]=={2,3,4};Select[Range[900000],f] -
PARI
list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\72)^(1/4), t1=p^4;forprime(q=2, (lim\t1)^(1/3), if(p==q, next);t2=t1*q^3;forprime(r=2, sqrt(lim\t2), if(p==r||q==r, next);listput(v,t2*r^2)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011
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Python
from math import isqrt from sympy import primepi, primerange, integer_nthroot def A190115(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(isqrt(x//(r**4*q**3))) for r in primerange(integer_nthroot(x,4)[0]+1) for q in primerange(integer_nthroot(x//r**4,3)[0]+1))+sum(primepi(integer_nthroot(x//p**5,4)[0]) for p in primerange(integer_nthroot(x,5)[0]+1))+sum(primepi(integer_nthroot(x//p**6,3)[0]) for p in primerange(integer_nthroot(x,6)[0]+1))+sum(primepi(isqrt(x//p**7)) for p in primerange(integer_nthroot(x,7)[0]+1))-(primepi(integer_nthroot(x,9)[0])<<1) return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025
Formula
Sum_{n>=1} 1/a(n) = P(2)*P(3)*P(4) - P(2)*P(7) - P(3)*P(6) - P(4)*P(5) + 2*P(9) = 0.00061171477910848082277..., where P is the prime zeta function. - Amiram Eldar, Mar 07 2024
Comments