A077313 -id:A077313 - cp's OEIS frontend

A077497 Primes of the form 2^r*5^s + 1.

2, 3, 5, 11, 17, 41, 101, 251, 257, 401, 641, 1601, 4001, 16001, 25601, 40961, 62501, 65537, 160001, 163841, 16384001, 26214401, 40960001, 62500001, 104857601, 167772161, 256000001, 409600001, 655360001, 2441406251, 2500000001, 4194304001, 10485760001

Offset: 1

Amarnath Murthy, Nov 07 2002

nonn

These are also the prime numbers p for which there is an integer solution x to the equation p*x = p*10^p + x, or equivalently, the prime numbers p for which (p*10^p)/(p-1) is an integer. - Vicente Izquierdo Gomez, Feb 20 2013

For n > 2, all terms are congruent to 5 (mod 6). - Muniru A Asiru, Sep 03 2017

			101 is in the sequence, since 101 = 2^2*5^2 + 1 and 101 is prime.

Ray Chandler, Table of n, a(n) for n = 1..3150

Cf. A005109, A077497, A077498, A077500, A003592, A077313, A019434.

K:=10^7;; # to get all terms <= K.
A:=Filtered(Filtered([1..K],i-> i mod 6=5),IsPrime);;
B:=List(A,i->Factors(i-1));;
C:=[];;  for i in B do if Elements(i)=[2] or Elements(i)=[2,5]  then Add(C,Position(B,i)); fi; od;
A077497:=Concatenation([2,3],List(C,i->A[i])); # Muniru A Asiru, Sep 03 2017

Do[p=Prime[k];s=FindInstance[p x == p 10^p+x,x,Integers];If[s!={},Print[p]],{k,10000}] (* Vicente Izquierdo Gomez, Feb 20 2013 *)

list(lim)=my(v=List(),t);for(r=0,log(lim)\log(5),t=5^r;while(t<=lim,if(isprime(t+1),listput(v,t+1)); t<<=1)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jan 29 2013

Corrected and extended by Reinhard Zumkeller, Nov 19 2002

More terms from Ray Chandler, Aug 02 2003

A077314 Primes of the form 2^r*7^s - 1.

Original entry on oeis.org

3, 7, 13, 31, 97, 127, 223, 1567, 3583, 4801, 6271, 8191, 19207, 25087, 33613, 76831, 131071, 401407, 524287, 917503, 1229311, 1605631, 3764767, 6588343, 14680063, 184473631, 737894527, 2147483647, 2259801991, 2877292543, 3758096383

Offset: 1

Amarnath Murthy, Nov 04 2002

nonn

In all terms except 3, r is odd. - Robert Israel, Jun 08 2018

Robert Israel, Table of n, a(n) for n = 1..1736

Cf. A023509, A077313.

N:= 10^30: # for all terms <= N
sort(select(isprime, [3, seq(seq(2^r*7^s-1, r=1..ilog2((N+1)/7^s),2),
s=0..floor(log[7]((N+1)/2)))])); # Robert Israel, Jun 08 2018

Corrected and extended by Ray Chandler, Aug 02 2003

A077315 Primes of the form 2^r * 11^s - 1.

Original entry on oeis.org

3, 7, 31, 43, 127, 241, 967, 5323, 8191, 117127, 131071, 524287, 7496191, 10307263, 77948683, 253755391, 428717761, 738197503, 1714871047, 2147483647, 16240345087, 27437936767, 42218553343, 1965081755647, 2414538435583, 7024111812607, 7860327022591, 16630113370111

Offset: 1

Amarnath Murthy, Nov 04 2002

nonn

Jinyuan Wang, Table of n, a(n) for n = 1..910

Cf. A023509, A077313, A077314, A173062.

lista(nn) = {my(q=1/2, p, w=List([])); for(r=0, logint(nn, 2), q=2*q; p=q/11; for(s=0, logint(nn\q, 11), p=11*p; if(ispseudoprime(p-1), listput(w, p-1)))); Set(w); } \\ Jinyuan Wang, Feb 23 2020

More terms from Ray Chandler, Aug 02 2003

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

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

Original entry on oeis.org

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

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

nonn

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.

			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), ...

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.

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

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

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

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

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

A086983 Primes of the form 2^r*p^s - 1, where p is an odd prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 31, 37, 43, 47, 53, 61, 67, 71, 73, 79, 97, 103, 107, 127, 151, 157, 163, 191, 193, 199, 211, 223, 241, 271, 277, 283, 313, 331, 337, 367, 383, 397, 421, 431, 457, 463, 487, 499, 523, 541, 547, 577, 607, 613, 631, 647, 661, 673

Offset: 1

Ray Chandler, Aug 02 2003

nonn

Primes p such that p+1 has at most one odd prime divisor.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A005105, A077497, A077498, A077499, A077500.

Cf. also A005109, A077313, A077314, A077315.

N:= 1000: # to get all terms <= N
Primes:= select(isprime, [$3..(N+1)/2]):
sort(convert(select(isprime, {2,seq(seq(seq(2^r*p^s-1, r = 1 .. ilog2((N+1)/p^s)),s=0..floor(log[p]((N+1)/2))),p=Primes)}),list)); # Robert Israel, Jun 13 2018

A292890 Primes of the form 2^r * 17^s - 1.

Original entry on oeis.org

3, 7, 31, 67, 127, 271, 577, 1087, 2311, 8191, 78607, 131071, 524287, 1114111, 2367487, 2672671, 17825791, 42762751, 90870847, 606076927, 2147483647, 5151653887, 5815734271, 9697230847, 329705848831, 474351505987, 700624928767, 892896952447, 1168231104511, 2482491097087

Offset: 1

Muniru A Asiru, Sep 26 2017

nonn

Primes of the forms 2^r * b^s - 1 where b = 1, 5, 7, 11, 13 are A000668 (Mersenne prime exponents), A077313, A077314, A077315 and A173062. When b = 3 we get A005105 with initial term 2.
For n > 1, all terms are congruent to 1 (mod 3).
Also, these are prime numbers p for which (34^p)/(p+1) is an integer.

			With n = 1, a(1) = 2^2 * 17^0 - 1 = 3.
With n = 4, a(4) = 2^2 * 17^1 - 1 = 67.
list of (r, s): (2, 0), (3, 0), (5, 0), (2, 1), (3, 1), (7, 0), (4, 1), (1, 2), (6, 1), (3, 2), (13, 0), (4, 3), (17, 0), (19, 0), (16, 1), (13, 2), (5, 4), (20, 1), (9, 4), (6, 5).

Cf. Sequences of primes of the forms 2^n * q^s - 1: A000668 (q = 1), A005105 (q = 3), A077313 (q = 5), A077314 (q = 7), A077315 (q = 11), A173062 (q = 13).

K:=10^7+1;; # to get all terms <= K.
A:=Filtered(Filtered([1..K], i->i mod 3=1),IsPrime);;    I:=[17];;
B:=List(A,i->Elements(Factors(i+1)));;
C:=List([0..Length(I)],j->List(Combinations(I,j),i->Concatenation([2],i)));;
A292890:=Concatenation([3],List(Set(Flat(List([1..Length(C)],i->List([1..Length(C[i])],j->Positions(B,C[i][j]))))),i->A[i]));

isok(p) = isprime(p) && (denominator((34^p)/(p+1)) == 1); \\ Michel Marcus, Sep 27 2017

More terms from Jinyuan Wang, Feb 23 2020

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A077497 Primes of the form 2^r*5^s + 1.

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A077314 Primes of the form 2^r*7^s - 1.

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A077315 Primes of the form 2^r * 11^s - 1.

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

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A086983 Primes of the form 2^r*p^s - 1, where p is an odd prime.

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A292890 Primes of the form 2^r * 17^s - 1.

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