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

1,1

Links

Table of n, a(n) for n=1..52.

Formula

-1 + A234102.

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

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A111173 A090793 A090791 * A026067 A039475 A377991

Adjacent sequences: A234096 A234097 A234098 * A234100 A234101 A234102

Keyword

nonn,easy

Author

Clark Kimberling, Dec 27 2013

Status

approved