A234102 Integers of the form (p*q*r + 1)/2, where p, q, r are distinct primes.

53, 83, 98, 116, 128, 137, 143, 173, 179, 193, 200, 215, 218, 228, 233, 242, 278, 281, 298, 305, 308, 314, 323, 326, 332, 333, 353, 358, 371, 380, 389, 398, 403, 431, 443, 449, 452, 458, 468, 479, 485, 494, 501, 503, 508, 512, 523, 533, 543, 548, 553, 557

Offset

1,1

Links

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

Formula

1 + A234099.

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

Example

53 = (3*5*7 + 1)/2.

Mathematica

t = Select[Range[1, 10000, 2], Map[Last, FactorInteger[#]] == Table[1, {3}] &]; Take[(t + 1)/2, 120] (* A234102 *)
v = Flatten[Position[PrimeQ[(t + 1)/2], True]] ; w = Table[t[[v[[n]]]], {n, 1, Length[v]}]  (* A234103 *)
(w + 1)/2  (* A234104 *)    (* Peter J. C. Moses, Dec 23 2013 *)

Prog

(Python)
from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A234102(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>>1 # Chai Wah Wu, Oct 18 2024

Crossrefs

Cf. A046389, A234099, A234103, A234104.

Sequence in context: A125876 A136065 A354915 * A234104 A212420 A001836

Adjacent sequences: A234099 A234100 A234101 * A234103 A234104 A234105

Keyword

nonn,easy

Author

Clark Kimberling, Dec 27 2013

Status

approved