A056725 -id:A056725 - cp's OEIS frontend

A245241 Integers n such that 6 * 7^n + 1 is prime.

0, 1, 4, 9, 99, 412, 2633, 5093, 5632, 28233, 36780, 47084, 53572

Offset: 1

Robert Price, Nov 14 2014

All terms correspond to verified primes, that is, not merely probable primes.

a(14) > 2*10^5.

			4 is in this sequence because 6 * 7^4 + 1 = 14407, which is prime.

Cf. A003307, A002235, A046865, A079906, A001771, A005541, A056725, A046867, A079907.

Cf. A005537, A005538, A005539, A216889, A216890, A247260.

Select[Range[0,200000], PrimeQ[6 * 7^# + 1] &]

A272057 Numbers n such that 3*4^n - 1 is prime.

Original entry on oeis.org

1, 2, 3, 9, 17, 19, 32, 38, 47, 103, 108, 153, 162, 229, 235, 637, 1638, 2102, 2567, 6338, 7449, 12845, 20814, 40165, 61815, 77965, 117380, 207420, 351019, 496350, 600523, 1156367, 2117707, 5742009, 5865925, 5947859

Offset: 1

Tim Johannes Ohrtmann, Apr 19 2016

These are Williams primes to base 3.

Half of the even terms of A002235.

H. C. Williams, The primality of certain integers of the form 2Ar^n - 1, Acta Arith. 39 (1981), 7-17.

Cf. A002235, A000043, A003307, A046865, A079906, A046866, A268061, A268356, A056725, A046867, A079907.

Select[Range[0, 100000], PrimeQ[3*4^# - 1] &]

for(n=1,10000, if(isprime(3*4^n-1), print1(n,", ")))

a(25) corrected and a(33)-a(36) added by Giovanni Resta, Apr 19 2016, using data from A002235.

A297348 Numbers k such that 12*13^k - 1 is prime.

Original entry on oeis.org

0, 2, 7, 11, 36, 164, 216, 302, 311, 455, 738, 1107, 2244, 3326, 4878, 8067, 46466

Offset: 1

Tim Johannes Ohrtmann, Feb 15 2018

Williams primes to base 12.

Steven Harvey, Williams primes
H. C. Williams, The primality of certain integers of the form 2Ar^n - 1, Acta Arith. 39 (1981), 7-17.

Cf. A003307, A272057, A046865, A079906, A046866, A268061, A268356, A056725, A046867, A079907, A273523.

[n: n in [0..10000] |IsPrime(12*13^n-1)];

Select[Range[0, 10000], PrimeQ[12*13^n-1] &]

for(n=0, 10000, if(isprime(12*13^n-1), print1(n, ", ")))

a(16) from Michael S. Branicky, Sep 14 2022

a(17) from Michael S. Branicky, Nov 12 2024

A305531 Smallest k >= 1 such that (n-1)*n^k + 1 is prime.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 3, 10, 3, 1, 2, 1, 1, 4, 1, 29, 14, 1, 1, 14, 2, 1, 2, 4, 1, 2, 4, 5, 12, 2, 1, 2, 2, 9, 16, 1, 2, 80, 1, 2, 4, 2, 3, 16, 2, 2, 2, 1, 15, 960, 15, 1, 4, 3, 1, 14, 1, 6, 20, 1, 3, 946, 6, 1, 18, 10, 1, 4, 1, 5, 42, 4, 1, 828, 1, 1, 2, 1, 12, 2, 6, 4, 30, 3, 3022, 2, 1, 1

Offset: 2

Eric Chen, Jun 04 2018

nonn

a(prime(j)) + 1 = A087139(j).
a(123) > 10^5, a(342) > 10^5, see the Barnes link for the Sierpinski base-123 and base-342 problems.
a(251) > 73000, see A087139.

Eric Chen, Table of n, a(n) for n = 2..122
Gary Barnes, Sierpinski conjectures and proofs
Eric Chen, Table n, a(n) for n = 2..360 status

For the numbers k such that these forms are prime:
a1(b): numbers k such that (b-1)*b^k-1 is prime
a2(b): numbers k such that (b-1)*b^k+1 is prime
a3(b): numbers k such that (b+1)*b^k-1 is prime
a4(b): numbers k such that (b+1)*b^k+1 is prime (no such k exists when b == 1 (mod 3))
a5(b): numbers k such that b^k-(b-1) is prime
a6(b): numbers k such that b^k+(b-1) is prime
a7(b): numbers k such that b^k-(b+1) is prime
a8(b): numbers k such that b^k+(b+1) is prime (no such k exists when b == 1 (mod 3)).
Using "-------" if there is currently no OEIS sequence and "xxxxxxx" if no such k exists (this occurs only for a4(b) and a8(b) for b == 1 (mod 3)):
.
   b   a1(b)   a2(b)   a3(b)   a4(b)   a5(b)   a6(b)   a7(b)   a8(b)
--------------------------------------------------------------------
   2 A000043 ------- A002235 A002253 A000043 ------- A050414 A057732
   3 A003307 A003306 A005540 A005537 A014224 A051783 A058959 A058958
   4 A272057 ------- ------- xxxxxxx A059266 A089437 A217348 xxxxxxx
   5 A046865 A204322 A257790 A143279 A059613 A124621 A165701 A089142
   6 A079906 A247260 ------- ------- A059614 A145106 A217352 A217351
   7 A046866 A245241 ------- xxxxxxx A191469 A217130 A217131 xxxxxxx
   8 A268061 A269544 ------- ------- A217380 A217381 A217383 A217382
   9 A268356 A056799 ------- ------- A177093 A217385 A217493 A217492
  10 A056725 A056797 A111391 xxxxxxx A095714 A088275 A092767 xxxxxxx
  11 A046867 A057462 ------- ------- ------- ------- ------- -------
  12 A079907 A251259 ------- ------- ------- A137654 ------- -------
  13 A297348 ------- ------- xxxxxxx ------- ------- ------- xxxxxxx
  14 A273523 ------- ------- ------- ------- ------- ------- -------
  15 ------- ------- ------- ------- ------- ------- ------- -------
  16 ------- ------- ------- xxxxxxx ------- ------- ------- xxxxxxx
Cf. (smallest k such that these forms are prime) A122396 (a1(b)+1 for prime b), A087139 (a2(b)+1 for prime b), A113516 (a5(b)), A076845 (a6(b)), A178250 (a7(b)).

a(n)=for(k=1,2^16,if(ispseudoprime((n-1)*n^k+1),return(k)))

A093948 Primes of the form 9*10^k - 1.

Original entry on oeis.org

89, 8999, 89999999, 89999999999999999999, 899999999999999999999999999999, 89999999999999999999999999999999999999, 8999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999

Offset: 1

Rick L. Shepherd, Apr 17 2004

nonn

Equivalently, primes of the form 8*10^k + 9*R_k, where R_k is the repunit (A002275) of length k.

Makoto Kamada, Prime numbers of the form 899...999.
Index entries for primes involving repunits.

Cf. A002275, A056725 (corresponding k).

Primes in A109002.

a(n) = 9*10^A056725(n) - 1 = A109002(A056725(n) + 1). - Elmo R. Oliveira, Jun 14 2025

A129990 Primes p such that the smallest integer whose sum of decimal digits is p is prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 19, 23, 29, 31, 41, 43, 71, 79, 97, 173, 179, 257, 269, 311, 389, 691, 4957, 8423, 11801, 14621, 25621, 26951, 38993, 75743, 102031, 191671, 668869

Offset: 1

J. M. Bergot, Jun 14 2007

			The smallest integer whose sum of digits is 17 is 89; 89 is prime, therefore 17 is in the sequence.

Cf. A051885.

Cf. A002957, A056703, A056712, A056716, A056721, A056725.

Select[Prime[Range[1000]],PrimeQ[FromDigits[Join[{Mod[ #,9]},Table[9,{i,1,Floor[ #/9]}]]]] &]

from itertools import islice
from sympy import isprime, nextprime
def A051885(n): return ((n%9)+1)*10**(n//9)-1 # from Chai Wah Wu
def agen(startp=2):
    p = startp
    while True:
        if isprime(A051885(p)): yield p
        p = nextprime(p)
print(list(islice(agen(), 23))) # Michael S. Branicky, Jul 27 2022

sorted( filter(is_prime, sum(([9*t+k for t in oeis(seq).first_terms()] for seq,k in (('A002957',1), ('A056703',2), ('A056712',4), ('A056716',5), ('A056721',7), ('A056725',8))), [3])) ) # Max Alekseyev, Feb 05 2025

Primes p such that (p mod 9 + 1) * 10^[p/9] - 1 is prime. Therefore the sequence consists of the term 3 and the primes of the forms A002957(k)*9+1, A056703(k)*9+2, A056712(k)*9+4, A056716(k)*9+5, A056721(k)*9+7, A056725(k)*9+8. - Max Alekseyev, Nov 09 2009

Edited, corrected and extended by Stefan Steinerberger, Jun 23 2007
Extended by D. S. McNeil, Mar 20 2009
a(29)-a(33) from Max Alekseyev, Nov 09 2009

cp's OEIS Frontend

A245241 Integers n such that 6 * 7^n + 1 is prime.

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A272057 Numbers n such that 3*4^n - 1 is prime.

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A297348 Numbers k such that 12*13^k - 1 is prime.

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A305531 Smallest k >= 1 such that (n-1)*n^k + 1 is prime.

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A093948 Primes of the form 9*10^k - 1.

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A129990 Primes p such that the smallest integer whose sum of decimal digits is p is prime.

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