A173062 Primes of the form 2^r * 13^s - 1.

3, 7, 31, 103, 127, 337, 1663, 5407, 8191, 131071, 346111, 524287, 2970343, 3655807, 22151167, 109051903, 617831551, 1631461441, 2007952543, 2147483647, 32127240703, 194664464383, 275716983697, 958348132351, 1357375919743, 1670616516607, 49834102882303, 57349132609183

Offset: 1

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Feb 09 2010

Keywords

• nonn

Comments

s = 0 is "trivial" case of Mersenne primes: 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ...

Mersenne prime exponents r: 2, 3, 5, 7, 13, 17, 19, 31, ...

Necessarily r odd as for r = 2*k and p a prime of form 6*n+1: 2^(2*k) * p^j - 1 a multiple of 3.

Proof by induction with 2^2 * p^1 - 1 = 4*(6*n+1) - 1 = 3*(8*n+1), 2^2(k+1) * p^j - 1 = 4* (2^k * p^j - 1) + 3.

No prime in case i = j = k (k>1) as a^k - 1 has divisor a - 1.

Examples

			2^2*13^0 - 1 = 3 = prime(2) => a(1).
2^3*13^1 - 1 = 103 = prime(27) => a(4).
2^7*13^9 - 1 = 1357375919743 = prime(50467169414) => a(25).
list of (r,s) pairs: (2,0), (3,0), (5,0), (3,1), (7,0), (1,2), (7,1), (5,2), (13,0), (17,0), (11,2), (19,0), (3,5), (7,4), (17,2), (23,1), (7,6), (1,8), (5,7), (31,0), (9,7), (19,5), (1,10), (25,4), (7,9), (11,8), (27,5), (5,11), (25,6), (19,8), (13,10), (3,13), (29,7), (5,14), (39,5), (15,13), (5,16), ...
		

References

• Peter Bundschuh, Einfuehrung in die Zahlentheorie, Springer-Verlag GmbH Berlin, 2002.
• Leonard E. Dickson, History of the Theory of numbers, vol. I, Dover Publications, 2005.
• Paulo Ribenboim, Wilfrid Keller, and Joerg Richstein, Die Welt der Primzahlen, Springer-Verlag GmbH Berlin, 2006.

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..950

Crossrefs

Cf. A000668, A005105, A077313, A077314, A077315.

Programs

PARI

lista(nn) = {my(q=1/2, p, w=List([])); for(r=0, logint(nn, 2), q=2*q; p=q/13; for(s=0, logint(nn\q, 13), p=13*p; if(ispseudoprime(p-1), listput(w, p-1)))); Set(w); } \\ Jinyuan Wang, Feb 23 2020

Python

from itertools import count, islice
from sympy import integer_log, isprime
def A173062_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 g(n):
        def f(x): return n+x-sum(((x+1)//13**i).bit_length() for i in range(integer_log(x+1,13)[0]+1))
        return bisection(f,n-1,n-1)
    return filter(lambda n:isprime(n), map(g,count(1)))
A173062_list = list(islice(A173062_gen(),30)) # Chai Wah Wu, Mar 31 2025

Extensions

a(26)-a(28) from Jinyuan Wang, Feb 23 2020