A123159 -id:A123159 - cp's OEIS frontend

A088783 Numbers n such that 10*n^k + 1 is composite for all k > 0.

12, 23, 32, 34, 45, 56, 65, 67, 78, 89, 98, 100, 111, 122, 131, 133, 144, 155, 164, 166, 177

Offset: 1

Ray Chandler, Oct 23 2003

All terms in the sequence fit the pattern 11m + 1 or 33m - 1 up to a(21) = 177.  Note that 10*(11m + 1)^k + 1 is divisible by 11 for all k, and 10*(33m - 1)^k + 1 is divisible by 3 when k is odd and 11 when k is even.
A prime 10*173^k+1 is now known (for k=264234, see link). The next blocker is 10*185^k+1 (is 10 a base-185 Sierpiński number?). - Jeppe Stig Nielsen, Apr 30 2018
10*185^k+1 is composite for all k <= 10^6 (see the Barnes link). - Eric Chen, Jun 07 2018
If a prime 10*185^k+1 were found, then this sequence would continue with 188, 197, 199, 210, 221, 230, 232, 243, 254, 263, 265, 276, 287, 296, 298, 309, 320, 329, 331, ... - Eric Chen, Jun 07 2018
The prime 10*173^264234+1 is found to solve the generalized Sierpinski problem in base 173 (with conjectured smallest Sierpinski number k=28, see the Barnes link and A123159 for these problems). - Eric Chen, Jun 07 2018
All unknown terms below 1024 are 185, 338, 417, 432, 537, 614, 668, 743, 744, 773, 786, 827, 863, 929, 935, 977, 986, 1000, 1004. Search limits: 185 at 10^6, 417 at 4*10^5, 743, 773 and 935 at 2*10^5, 338, 744 and 977 at 10^5, 432 at 25000, other numbers except 1000 at 5000, 1000 is corresponding the generalized Fermat prime in base 10 and already searched to (2^24-1)/3, since the smallest prime of the form 10^n+1 greater than 101 is at least 10^(2^24)+1. - Eric Chen, Jun 09 2018
Large primes with n <= 1024 and exponent > 10^4: 10*173^264234+1, 10*198^47664+1, 10*311^314806+1, 10*341^106008+1, 10*449^18506+1, 10*492^42842+1, 10*605^12394+1, 10*708^17562+1, 10*710^31038+1, 10*800^15104+1, 10*802^149319+1, 10*879^25003+1, they are all proven primes, i.e., not merely probable primes, since they can be proved prime with the N-1 method. - Eric Chen, Jun 09 2018
All other n <= 1024 with n != 1 (mod 11) and n != 32 (mod 33) have at least one prime of the form 10*n^k+1 with k <= 10^4. - Eric Chen, Jun 09 2018

Gary Barnes, Sierpinski conjectures and proofs
Chris K. Caldwell, The Prime Database: 10*173^264234 + 1

Cf. Indices of zero entries in A088622 & A088782.

for(n=2,10^3,if(n%11==1||n%33==32,print1(n,", ");next());for(k=1,+oo,ispseudoprime(10*n^k+1)&&next(2))) \\ Jeppe Stig Nielsen, Apr 30 2018

n = 11m+1 and n = 33m-1 for m > 0.

a(21) from Jeppe Stig Nielsen, Apr 30 2018

A250204 Sierpiński problem in base 6: Least k > 0 such that n*6^k+1 is prime, or 0 if no such k exists.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 4, 0, 1, 1, 1, 1, 0, 2, 1, 1, 1, 0, 5, 1, 4, 1, 0, 1, 1, 1, 2, 0, 1, 2, 1, 1, 0, 1, 2, 1, 1, 0, 1, 5, 5, 2, 0, 1, 1, 1, 3, 0, 2, 1, 1, 7, 0, 1, 1, 2, 1, 0, 2, 1, 1, 1, 0, 2, 1, 8, 1, 0, 1, 2, 1, 1, 0, 7, 1, 1, 4, 0, 4, 1, 2, 1, 0, 2, 5, 1, 2, 0, 1, 1, 2, 3, 0, 1, 1, 9, 2, 0, 1, 1, 1, 1, 0, 1, 6, 1, 2, 0, 1, 3, 1, 4, 0, 1, 2, 23, 1, 0, 4

Offset: 1

Eric Chen, Mar 11 2015

nonn

a(5k+4) = 0, since (5k+4)*6^n+1 is always divisible by 5, but there are infinitely many numbers not in the form 5k+4 such that a(n) = 0. For example, a(174308) = 0 since 174308*6^n+1 is always divisible by 7, 13, 31, 37, or 97 (See A123159). Conjecture: if n is not in the form 5k+4 and n < 174308, then a(n) > 0.

However, according to the Barnes link no primes n*6^k+1 are known for n = 1296, 7776 and 46656, so these may be counterexamples. - Robert Israel, Mar 17 2015

Eric Chen, Table of n, a(n) for n = 1..1000
Gary Barnes, Sierpinski conjectures and proofs

Cf. A040076, A046067, A078680, A033809, A123159.

Cf. A250205 (Least k > 0 such that n*6^k-1 is prime).

N:= 1000: # to get a(1) to a(N), using k up to 10000
a[1]:= 1:
for n from 2 to N do
  if n mod 5 = 4 then a[n]:= 0
  else
    for k from 1 to 10000 do
    if isprime(n*6^k+1) then
       a[n]:= k;
       break
    fi
    od
  fi
od:
L:= [seq(a[n],n=1..N)]; # Robert Israel, Mar 17 2015

(* m <= 10000 is sufficient up to n = 1000 *)
a[n_] := For[k = 1, k <= 10000, k++, If[PrimeQ[n*6^k + 1], Return[k]]] /. Null -> 0; Table[a[n], {n, 1, 120}]

a(n) = if(n%5==4, 0, for(k = 1, 10000, if(ispseudoprime(n*6^k+1), return(k))))

A251057 Odd numbers n not congruent to 5 mod 6 such that for all k >= 1 the numbers n*4^k + 1 are composite.

Original entry on oeis.org

66741, 271129, 308481, 327739, 436029, 482719, 575041, 636921, 934909, 965431, 1259541, 1259779, 1384059, 1518781, 1639459, 1997589, 2038371, 2131099, 2191531, 2397951, 2473929, 2541601, 2576089, 2931991, 2965569, 3098059, 3608251, 3885579, 3999399, 4095859

Offset: 1

Arkadiusz Wesolowski, Dec 15 2014

nonn

A123159(4) = a(1).

a(n) is a Sierpiński number if it is of the form 6*m + 1.

Cf. A076336, A123159, A251757.

A291437 Smallest m >= 0 such that (2n)3^m + 1 is prime, or -1 if no such value exists.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 4, 0, 0, 2, 0, 2, 1, 0, 1, 9, 0, 0, 4, 1, 0, 2, 0, 0, 1, 1, 0, 1, 0, 2, 4, 0, 1, 1, 1, 0, 2, 0, 0, 1, 0, 0, 1, 0, 3, 1, 2, 4, 1, 1, 0, 2, 0, 1, 5, 0, 0, 1, 2, 1, 1, 0, 0, 1, 1, 0, 2, 80, 0, 6, 0, 8, 2, 0, 1

Offset: 1

Martin Renner, Aug 23 2017

nonn

There exist even integers 2*n such that (2*n)*3^m + 1 is always composite.
It is conjectured that the smallest one is 125050976086 = A123159(3), therefore a(62525488043) = -1.
For the corresponding primes see A291438.
a(A005097(n)) = 0 and a(A047845(n+1)) > 0 (or = -1).

			a(4) = 2 because this is the smallest value such that 8*3^2 + 1 = 73 is prime, since 8*3^0 + 1 = 9 and 8*3^1 + 1 = 25 are not prime.

Cf. A005097, A046067, A047845, A123159, A291438.

a:=[]:
for n from 1 to 10^3 do
  t:=-1:
  for m from 0 to 10^3 do # this max value of m is sufficient up to n=10^3
    if isprime((2*n)*3^m+1) then t:=m: break: fi:
  od:
  a:=[op(a),t]:
od:
a;

Table[SelectFirst[Range[0, 10^3], PrimeQ[2 n*3^# + 1] &] /. k_ /; MissingQ@ k -> -1, {n, 104}] (* Michael De Vlieger, Aug 23 2017 *)

a(n) = {my(m = 0); while (!isprime(p=(2*n)*3^m + 1), m++); m;} \\ Michel Marcus, Aug 25 2017

A291438 Smallest prime of the form (2n)3^m + 1 for some m >= 0, or -1 if no such prime exists.

Original entry on oeis.org

3, 5, 7, 73, 11, 13, 43, 17, 19, 61, 23, 73, 79, 29, 31, 97, 103, 37, 3079, 41, 43, 397, 47, 433, 151, 53, 163, 1102249, 59, 61, 5023, 193, 67, 613, 71, 73, 223, 229, 79, 241, 83, 757, 6967, 89, 271, 277, 283, 97, 883, 101, 103, 313, 107, 109, 331, 113, 3079

Offset: 1

Martin Renner, Aug 23 2017

nonn

There exist even integers 2*n such that (2*n)*3^m + 1 is always composite.
It is conjectured that the smallest one is 125050976086 = A123159(3), therefore a(62525488043) = -1.
For the corresponding numbers m see A291437.

			a(4) = 73 because 8*3^2 + 1 = 73 is the smallest prime of this form, since 8*3^0 + 1 = 9 and 8*3^1 + 1 = 25 are not prime.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Cf. A057025, A123159, A291437.

a:=[]:
for n from 1 to 10^3 do
  t:=-1:
  for m from 0 to 10^3 do # this max value of m is sufficient up to n=10^3
    if isprime((2*n)*3^m+1) then t:=m: break: fi:
  od:
  a:=[op(a),(2*n)*3^t+1]:
od:
a;

Table[If[# < 0, #, 1 + 2 n*3^#] &@ SelectFirst[Range[0, 10^3], PrimeQ[2 n*3^# + 1] &] /. k_ /; MissingQ@ k -> -1, {n, 60}] (* Michael De Vlieger, Aug 23 2017 *)

a(n) = {my(m = 0); while (!isprime(p=(2*n)*3^m + 1), m++); p;} \\ Michel Marcus, Aug 25 2017

A257645 a(n) = 15*n + 14.

Original entry on oeis.org

14, 29, 44, 59, 74, 89, 104, 119, 134, 149, 164, 179, 194, 209, 224, 239, 254, 269, 284, 299, 314, 329, 344, 359, 374, 389, 404, 419, 434, 449, 464, 479, 494, 509, 524, 539, 554, 569, 584, 599, 614, 629, 644, 659, 674, 689, 704, 719, 734, 749, 764, 779

Offset: 0

Arkadiusz Wesolowski, Nov 05 2015

A123159(a(n)) <= 4.
This is not a subsequence of A047725 (for example, 239 is missing in A047725). - Bruno Berselli, Nov 06 2015
Equivalently, intersection of A016897 and A016789. - Bruno Berselli, Jan 24 2018

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008597, A016897, A175887, A123159, A204542.

Magma
```
[15*n+14: n in [0..51]];
```
Maple
```
seq(15*n+14, n=0..51);
```
Mathematica
```
15 Range[50] - 1
```
PARI
```
for(n=0, 51, print1(15*n+14, ", "));
```

G.f.: (14 + x)/(1 - x)^2.
a(n) = A008597(n+1) - 1. - Omar E. Pol, Nov 05 2015
a(n) = A016897(3n+2) = A175887(2n+2) = A204542(4n+4). - Bruno Berselli, Nov 06 2015
E.g.f.: (15*x + 14)*exp(x). - G. C. Greubel, Apr 23 2018
a(n) = 2*a(n-1)-a(n-2). - Wesley Ivan Hurt, Dec 27 2023

A263500 Smallest positive b such that for all k >= 1 the numbers n*b^k + 1 are composite and gcd(n+1, b-1) = 1, or 0 if n > 1 is of the form 2^m - 1.

Original entry on oeis.org

0, 14, 140324348, 34, 0, 8, 177744, 32, 14

Offset: 3

Arkadiusz Wesolowski, Oct 19 2015

a(2) = 131072, or it is a Sierpiński number multiplied by a power of 2.
Because a(1) = 8 but a(2) is unknown, we do not include it here; we set the offset to 3.
a(2) <= 201446503145165177.

Dan Krywaruczenko, A Reverse Sierpiński Number Problem, Rose-Hulman Undergraduate Mathematics Journal: Vol. 9: Iss. 2, Article 4, 2009.

Cf. A123159.

A283619 a(n) = (conjectured) smallest positive integer k which is neither of the form p + n^x nor of the form p - n^x with x >= 0 and p prime, where gcd(k, n) = 1 and gcd(k^2-1, n-1) = 1.

Original entry on oeis.org

30666137, 3902132276156, 2473929, 1015214, 464437, 40743218950116, 47, 2344, 61863, 32660, 4367, 7974, 11, 2021170066180678, 92343, 784, 571, 2364594, 13, 20450, 136113, 2596, 176011, 262638, 3223, 512, 59217, 26, 18973, 6360528, 23, 11848, 99, 292226, 832573

Offset: 2

Arkadiusz Wesolowski, Mar 12 2017

nonn

The definition is similar to that for A123159, but considering "p + n^x" and "p - n^x".
What does "conjectured" mean? A positive integer k is a candidate if:
1) gcd(k, n) = 1,
2) gcd(k^2-1, n-1) = 1,
3) every term in the sequence k + n^x is divisible by one of the prime numbers of a covering set,
4) all numbers of the form k - n^x are composite, k > n^x + 1, x >= 0.
The main problem is to prove that the given terms are indeed correct.
A quick search showed that a(8) = 47, a(14) = 11, a(20) = 13, a(27) = 512, a(29) = 26, a(32) = 23, a(34) = 99.
This is an interesting sequence: it leads to new classes of numbers. For example, the integer 30666137 is probably the smallest number that is simultaneously a Polignac number and a Sierpinski number.

Cf. A076336, A123159, A263644, A283622.

A345698 Sierpiński problem in base 5: a(n) is the smallest k >= 0 such that (2n)5^k + 1 is prime, or -1 if no such k exists.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 3, 8, 0, 1, 0, 0, 3, 0, 1, 1, 0, 1, 1, 0, 0, 1, 2, 0, 3, 0, 0, 257, 2, 0, 1, 0, 1, 1, 0, 2, 1, 2, 0, 1, 0, 0, 1, 0, 0, 3, 0, 1, 15, 4, 1, 79, 48, 0, 1, 0, 1, 5, 0, 0, 1, 6, 4, 3, 0, 0, 1, 2, 0, 3, 2, 0, 1, 0, 2, 7

Offset: 1

Felix Fröhlich, Jun 24 2021

sign

a(159986/2) = a(79993) = -1.

			For n = 17: 34*5^k + 1 is composite for k = 0, 1, 2, 3, 4, 5, 6, 7 and prime for k = 8. Since 8 is the smallest such k, a(17) = 8.

Joe O, Project Description, Mersenne forum.
Reggie, Welcome to the Sierpinski/Riesel Base 5 Project, PrimeGrid forum.
Wikipedia, Sierpiński number

Cf. A123159, A291437 (Sierpiński problem base 3), A345403 (Riesel problem base 5).

a(n) = for(k=0, oo, if(ispseudoprime((2*n)*5^k+1), return(k)))

cp's OEIS Frontend

A088783 Numbers n such that 10*n^k + 1 is composite for all k > 0.

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A250204 Sierpiński problem in base 6: Least k > 0 such that n*6^k+1 is prime, or 0 if no such k exists.

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A251057 Odd numbers n not congruent to 5 mod 6 such that for all k >= 1 the numbers n*4^k + 1 are composite.

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A291437 Smallest m >= 0 such that (2*n)*3^m + 1 is prime, or -1 if no such value exists.

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A291438 Smallest prime of the form (2*n)*3^m + 1 for some m >= 0, or -1 if no such prime exists.

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A257645 a(n) = 15*n + 14.

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A263500 Smallest positive b such that for all k >= 1 the numbers n*b^k + 1 are composite and gcd(n+1, b-1) = 1, or 0 if n > 1 is of the form 2^m - 1.

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A283619 a(n) = (conjectured) smallest positive integer k which is neither of the form p + n^x nor of the form p - n^x with x >= 0 and p prime, where gcd(k, n) = 1 and gcd(k^2-1, n-1) = 1.

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A345698 Sierpiński problem in base 5: a(n) is the smallest k >= 0 such that (2*n)*5^k + 1 is prime, or -1 if no such k exists.

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A291437 Smallest m >= 0 such that (2n)3^m + 1 is prime, or -1 if no such value exists.

A291438 Smallest prime of the form (2n)3^m + 1 for some m >= 0, or -1 if no such prime exists.

A345698 Sierpiński problem in base 5: a(n) is the smallest k >= 0 such that (2n)5^k + 1 is prime, or -1 if no such k exists.