A088622 -id:A088622 - cp's OEIS frontend

A088782 a(n) is the smallest k>0 such that concatenation of n^k and 1 is prime, or 0 if no such number exists.

1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 10, 0, 1, 6, 1, 3, 1356, 1, 1, 2, 1, 15, 0, 1, 1, 2, 1, 1, 4, 2, 1, 0, 1, 0, 2, 3, 2, 4, 4, 1, 2, 1, 1, 6, 0, 1, 2, 2, 1, 4, 3, 1, 16, 1, 9, 0, 1, 2, 36, 1, 165, 66, 1, 1, 0, 1, 0, 6, 1, 1, 2, 12, 3, 138, 1, 1, 4, 0, 5, 4, 1, 1, 2, 5, 2, 2, 3, 1, 0, 2, 1, 24, 2, 1, 42, 7, 1, 0, 1

Offset: 1

Ray Chandler, Oct 23 2003

a(185) > 10^6, see link. - Eric Chen, Jul 16 2019

Eric Chen, Table of n, a(n) for n = 1..184
Karsten Bonath, Proth prime small bases least n
Eric Chen, Table of n, a(n) for n = 1..500 status

Cf. A088622, A088783.

f[n_] := Block[{k = 1}, While[ !PrimeQ[10*n^k + 1], k++ ]; k]; f[1] = 1; Table[ f[n], {n, 1, 11}] (* Robert G. Wilson v, Oct 29 2003 *)

a(n)=if((n%11==1 || n%33==32) && n>1, 0, for(k=1, 10^6, if(ispseudoprime(10*n^k+1), return(k)))) \\ Eric Chen, Jul 16 2019

A088623 Prime obtained as the concatenation 1 followed by the smallest power of n, or 0 if no such number exists.

Original entry on oeis.org

11, 0, 13, 0, 0, 0, 17, 0, 19, 0, 114641, 0, 113, 0, 0, 0, 1289, 0, 1361, 0, 11025506433613486607375777617584133309366191904729927960524981845743709132117581, 0, 1907846434775996175406740561329, 0, 0, 0, 127, 0, 1500246412961, 0, 131, 0

Offset: 1

Amarnath Murthy, Oct 19 2003

Cf. A088622.

f[n_] := Block[{k = 1}, While[ p = ToExpression["1" <> ToString[n^k]]; !PrimeQ[p], k++ ]; p]; g[n_] := If[ Mod[n, 10] == 1 || Mod[n, 10] == 3 || Mod[n, 10] == 7 || Mod[n, 10] == 9, f[n], 0]; Table[ g[n], {n, 1, 33}] (* Robert G. Wilson v, Oct 31 2003 *)

a(2k)=a(5k)=0. - Ray Chandler, Oct 23 2003

Corrected and extended by Ray Chandler, Oct 23 2003

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

Original entry on oeis.org

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

A089776 If n mod 10 is 1, 3, 7, or 9, then a(n) = least prime of the form 1 followed by n^r; else a(n) = least prime of the form n^r followed by a 1. In both cases r must be > 1 and a(n) = 0 if no such prime exists.

Original entry on oeis.org

11, 41, 13, 41, 251, 61, 17, 641, 19, 101, 114641, 0, 113, 75295361, 151, 40961, 1289, 181, 1361, 4001, 11025506433613486607375777617584133309366191904729927960524981845743709132117581, 1368800680154120519681, 1907846434775996175406740561329, 241, 251, 6761, 127, 281, 1500246412961, 9001, 131

Offset: 1

Amarnath Murthy, Nov 24 2003

Subsidiary sequence: 3 more sequences can be obtained by replacing 1 by 3,7 and 9.

			a(12) = 0 because 1+10*12^r is always divisible by 11.
a(32) = 0 because 1+10*32^r is divisible by 3 if r is odd and by 11 if r is even.

Cf. A088622, A088623.

If n is divisible by 2 or 5, a(n) = A088622(n); otherwise a(n) = A088623(n). - David Wasserman, Oct 12 2005

More terms from David Wasserman, Oct 12 2005

A267720 The smallest prime number formed by the concatenation of consecutive powers of n beginning with 1^0 = "1," and ending with a trailing "1" - or "0," if no such prime number exists.

Original entry on oeis.org

0, 131, 1416641, 151, 16361, 17491, 181, 191, 0, 1111211, 11214417282073624883229859841, 0, 11419627441, 1151, 1162564096655361048576167772162684354564294967296687194767361099511627776175921860444161, 1171, 1181, 11936168591, 1201

Offset: 2

Thomas S. Pedigo, Jan 19 2016

It is not known whether solutions exist for A(2), A(10), or A(13); if there are any, they must be greater than 10^128.
A(10) has final exponent 35; "110100100010000...1000000000000000000000000000000000001", total length 667 digits. - Dana Jacobsen, Jan 22 2016
A(2) has exponent > 1100, length > 182841 digits.  A(13) has exponent > 750, length > 314093 digits. - Dana Jacobsen, Jan 25 2016

			A(14)=11419627441; 14^0=1; 14^1=14; 14^2=196; 14^3=2744; 11419627441 is prime.
A(19)=11936168591; 19^0=1; 19^1=19; 19^2=361; 19^3=6859; 11936168591 is prime.

Cf. A068704, A088622.

lim = 60; Table[SelectFirst[Rest@ FoldList[FromDigits@ Join[Most@ IntegerDigits@ #1, IntegerDigits[n^#2], {1}] &, 11, Range@ lim], PrimeQ], {n, 2, 20}] /. k_ /; MissingQ@ k -> 0 (* Michael De Vlieger, Jan 25 2016, Version 10.2 *)

a(8) corrected by Thomas S. Pedigo, Dec 08 2021

cp's OEIS Frontend

A088782 a(n) is the smallest k>0 such that concatenation of n^k and 1 is prime, or 0 if no such number exists.

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A088623 Prime obtained as the concatenation 1 followed by the smallest power of n, or 0 if no such number exists.

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

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A089776 If n mod 10 is 1, 3, 7, or 9, then a(n) = least prime of the form 1 followed by n^r; else a(n) = least prime of the form n^r followed by a 1. In both cases r must be > 1 and a(n) = 0 if no such prime exists.

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A267720 The smallest prime number formed by the concatenation of consecutive powers of n beginning with 1^0 = "1," and ending with a trailing "1" - or "0," if no such prime number exists.

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