A087053 Numbers of the form pq + qr + rp where p, q and r are distinct primes, with multiplicity.

31, 41, 61, 59, 71, 91, 71, 87, 101, 101, 121, 113, 103, 129, 151, 131, 161, 143, 119, 191, 171, 131, 167, 211, 151, 221, 185, 151, 241, 167, 191, 213, 227, 271, 221, 199, 301, 191, 311, 269, 243, 167, 211, 341, 275, 297, 269, 361, 215, 311, 293, 247, 371

Offset: 1

Author

Reinhard Zumkeller, Aug 07 2003

Keywords

• nonn

Comments

Arithmetic derivative of numbers having exactly three primes that are distinct: a(n) = A003415(A007304(n)).

Crossrefs

Cf. A087054, A238397.

Programs

PARI

is(n)=forprime(r=(sqrtint(3*n-3)+5)\3, (n-6)\5, forprime(q= sqrtint(r^2+n)-r+1, min((n-2*r)\(r+2), r-2), if((n-q*r)%(q+r)==0 && isprime((n-q*r)/(q+r)), return(1)))); 0 \\ Charles R Greathouse IV, Feb 26 2014

PARI

list(n)=my(v=List()); forprime(r=5, (n-6)\5, forprime(q=3, min((n-2*r)\(r+2), r-2), my(S=q+r, P=q*r); forprime(p=2, min((n-P)\S, q-1), listput(v, p*S+P))));  Set(v) \\ Charles R Greathouse IV, Feb 26 2014

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot, primefactors
def A087053(n):
    def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1),1) for b,m in enumerate(primerange(k+1,isqrt(x//k)+1),a+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return (p:=primefactors(bisection(f)))[0]*(p[1]+p[2])+p[1]*p[2] # Chai Wah Wu, Aug 30 2024