A347977 Primes of the form 2^p * 3^q * 5^r * 7^s - 1.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 53, 59, 71, 79, 83, 89, 97, 107, 127, 139, 149, 167, 179, 191, 199, 223, 239, 251, 269, 293, 349, 359, 383, 419, 431, 449, 479, 499, 503, 587, 599, 647, 719, 809, 839, 863, 881, 971, 1049, 1151, 1249, 1259, 1279, 1399, 1439, 1499, 1511, 1567, 1619, 1889

Offset: 1

Author

Flávio V. Fernandes, Sep 21 2021

Keywords

• nonn

Comments

Restricting to r = s = 0 gives A005105; s = 0 gives A293194.

Primes of the form A002473(k) - 1.

Examples

			251 = 2^2 * 3^2 * 5^0 * 7^1 - 1 and 839 = 2^3 * 3^1 * 5^1 * 7^1 - 1 are terms.
		

Links

• Flávio V. Fernandes, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002473, A005105, A293194.

Programs

Mathematica

With[{n = 2000}, Sort@ Select[Flatten@ Table[2^p * 3^q * 5^r * 7^s - 1, {p, 0, Log[2, n]}, {q, 0, Log[3, n/(2^p)]}, {r, 0, Log[5, n/(2^p * 3^q)]}, {s, 0, Log[7, n/(2^p * 3^q * 5^r)]}], PrimeQ]] (* Amiram Eldar, Sep 25 2021 after Michael De Vlieger at A293194 *)

PARI

isok(p) = isprime(p) && (vecmax(factor(p+1)[,1]) < 11); \\ Michel Marcus, Nov 10 2021

PARI

upto(limit)={my(P=[2,3,5,7]); local(L=List()); my(recurse(k,t) = if(t<=limit+1, if(k>#P, if(isprime(t-1), listput(L,t-1)), my(b=P[k]); for(e=0, logint(limit+1,b), self()(k+1, t*b^e))))); recurse(1, 1); vecsort(Vec(L))} \\ Andrew Howroyd, Nov 20 2021

Python

from itertools import count, islice
from sympy import integer_log, isprime
def A347977_gen(): # generator of terms
    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):
        c = x
        for i in range(integer_log(x,7)[0]+1):
            for j in range(integer_log(m:=x//7**i,5)[0]+1):
                for k in range(integer_log(r:=m//5**j,3)[0]+1):
                    c -= (r//3**k).bit_length()
        return c
    yield from filter(isprime,(bisection(lambda k:n+f(k),n,n)-1 for n in count(1)))
A347977_list = list(islice(A347977_gen(),30)) # Chai Wah Wu, Mar 31 2025