A376699 Positions of primes in the sequence of numbers of the form 2^i * 3^j - 1 (A069353).

3, 4, 5, 6, 8, 10, 11, 13, 15, 16, 18, 21, 22, 25, 31, 32, 36, 39, 40, 42, 51, 57, 61, 63, 65, 66, 71, 73, 79, 82, 94, 97, 106, 107, 110, 120, 121, 127, 128, 129, 130, 138, 142, 144, 161, 192, 204, 205, 212, 216, 232, 234, 244, 259, 264, 265, 308, 329, 346, 348

Offset: 1

Author

Amiram Eldar, Oct 02 2024

Keywords

• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A003586, A005105, A069353, A375906, A376700, A376701.

Programs

Mathematica

With[{lim = 10^10}, Position[Sort@ Flatten@ Table[2^i*3^j - 1, {i, 0, Log2[lim]}, {j, 0, Log[3, lim/2^i]}], _?PrimeQ] // Flatten]

PARI

lista(lim) = {my(s = List()); for(i = 0, logint(lim, 2), for(j = 0, logint(lim >> i, 3), listput(s, 2^i * 3^j - 1))); s = Set(s); for(i = 1, #s, if(isprime(s[i]), print1(i, ", ")));}

Python

from itertools import count, islice
from sympy import isprime, integer_log
def A069353(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(((x+1)//3**i).bit_length() for i in range(integer_log(x+1,3)[0]+1))
    return bisection(f,n-1,n-1)
def A376699_gen(): # generator of terms
    return filter(lambda n:isprime(A069353(n)), count(1))
A376699_list = list(islice(A376699_gen(),30)) # Chai Wah Wu, Mar 31 2025

Formula

A069353(a(n)) = A003586(a(n)) - 1 = A005105(n).