A234099 Integers of the form (p*q*r - 1)/2, where p, q, r are distinct primes.

52, 82, 97, 115, 127, 136, 142, 172, 178, 192, 199, 214, 217, 227, 232, 241, 277, 280, 297, 304, 307, 313, 322, 325, 331, 332, 352, 357, 370, 379, 388, 397, 402, 430, 442, 448, 451, 457, 467, 478, 484, 493, 500, 502, 507, 511, 522, 532, 542, 547, 552, 556

Offset: 1

Author

Clark Kimberling, Dec 27 2013

Keywords

• nonn
• easy

Examples

			52 = (3*5*7 - 1)/2.
		

Crossrefs

Cf. A046389, A234100, A234101, A234093, A234102.

Programs

Mathematica

t = Select[Range[1, 10000, 2], Map[Last, FactorInteger[#]] == Table[1, {3}] &]; Take[(t - 1)/2, 120] (* A234099 *)
v = Flatten[Position[PrimeQ[(t - 1)/2], True]] ; w = Table[t[[v[[n]]]], {n, 1, Length[v]}]  (* A234100 *)
(w - 1)/2 (* A234101 *)    (* Peter J. C. Moses, Dec 23 2013 *)

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A234099(n):
    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
    def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(3,integer_nthroot(x,3)[0]+1),2) for b,m in enumerate(primerange(k+1,isqrt(x//k)+1),a+1)))
    return bisection(f,n,n)>>1 # Chai Wah Wu, Oct 18 2024

Formula

-1 + A234102.

a(n) = (A046389(n)-1)/2. - Chai Wah Wu, Oct 18 2024