A171376 -id:A171376 - cp's OEIS frontend

A100028 Values of n for which the decimal number 10...030...01 is an n-digit prime.

3, 5, 7, 9, 23, 29, 33, 185, 267, 307, 757, 897, 1571, 2977, 3831, 4595, 6573, 9511, 11651, 15641, 68885, 69883

Offset: 1

Harvey Dubner (harvey(AT)dubner.com), Nov 20 2004

			The corresponding primes are 131, 10301, 1003001, 100030001, 10000000000300000000001, etc.

Cf. A100026, A100027, A100456, A100457, A100458, A100459.

IntegerLength/@Select[Table[FromDigits[Join[PadRight[{1},n,0],{3},PadLeft[ {1},n,0]]],{n,35000}],PrimeQ] (* Harvey P. Dale, Dec 20 2019 *)

a(n) = 2*(A171376(n+1))+1. - Chai Wah Wu, Aug 20 2015

A070247 Palindromic primes with digit sum 5.

Original entry on oeis.org

5, 131, 10301, 1003001, 100030001, 100111001, 101000010000101, 10000010101000001, 101000000010000000101, 110000000010000000011, 10000000000300000000001, 10000100000100000100001, 100000100000010000001000001, 10000000000000300000000000001, 10000000001000100010000000001

Offset: 1

Amarnath Murthy, May 05 2002

It is conjectured that are just 3 palindromic primes with digit sum 2, namely 2, 11 and 101. If any others exist, they must be of the form 10^(2^k) + 1 with k > 14.
From Jeppe Stig Nielsen, Aug 30 2025: (Start)
It is now known that any additional primes 10^(2^k) + 1 must have k >= 31.
Digit sum 3 yields only one prime, 3, a palindrome in a vacuous way.
Digit sum 4 leads to primes (A062339), but such numbers can never be palindromes. Proof: Let w be any palindrome with digit sum 4. So w = 10^a + 10^b + 10^c + 10^d with a >= b >= c >= d >= 0. But then 10^c + 10^d is a nontrivial divisor of w, showing that w is not prime.
You may have come here searching for the subsequence 5, 131, 10301, 1003001, 100030001, 10000000000300000000001, ... where the largest digit exceeds 1. See A171376 and A100028 for information on them.
(End)

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..386 (all terms below 10^1000; terms n = 1..238 from Chai Wah Wu)
Hans Riesel, Some factors of the numbers Gn = 6^2^n+1 and Hn = 10^2^n+1, Math. Comp. 23 (1969), p. 413-415. With errata reported in Math. Comp. 24 (1970), p. 243.

Cf. A002385, A062341, A070248, A070249.

Do[p = Join[ IntegerDigits[n, 4], Reverse[ Drop[ IntegerDigits[n, 4], -1]]]; q = Plus @@ p; If[q == 5 && PrimeQ[ FromDigits[p]] && q == 5, Print[ FromDigits[p]]], {n, 1, 4 10^8}] (* this coding will not pick up the first entry *)

for(i=0,50,for(j=0,i,p=10^(2*i)+10^(i+j)+10^i+10^(i-j)+1;isprime(p)&&print1(p,", "))) \\ Jeppe Stig Nielsen, Aug 30 2025

Edited by Robert G. Wilson v, May 15 2002

More terms from Chai Wah Wu, Nov 25 2015

A171411 Numbers k such that 1 + 5*10^k + 100^k is prime.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 27, 165, 230, 237, 369, 485, 628, 875, 964, 1419, 4083

Offset: 1

Jason Earls, Dec 08 2009

No more terms up to k=7600. The value corresponding to 4083 was certified prime with WinPFGW.

a(18) > 30000. - Michael S. Branicky, May 15 2025

Chris K. Caldwell, Prime Pages, Database Search Output

Cf. A171376, A171459, A171514, A171554.

is(n)=ispseudoprime(1+5*10^n+100^n) \\ Charles R Greathouse IV, Jun 13 2017

A171459 Numbers k such that 1 + 6*10^k + 100^k is prime.

Original entry on oeis.org

2, 4, 24, 32, 34, 72, 75, 164, 532, 1335, 4704, 29762

Offset: 1

Jason Earls, Dec 09 2009

No more terms up to k=7800. The value corresponding to 4704 was certified prime with WinPFGW.

Chris K. Caldwell, Prime Pages, Database Search Output

Cf. A171376, A171411, A171514, A171554.

is(n)=ispseudoprime(1+6*10^n+100^n) \\ Charles R Greathouse IV, Jun 13 2017

a(12) using Caldwell link from Michael S. Branicky, Jun 30 2024

A171514 Numbers k such that 1 + 8*10^k + 100^k is prime.

Original entry on oeis.org

1, 3, 6, 9, 13, 17, 29, 63, 90, 531, 14286, 30617, 37815

Offset: 1

Jason Earls, Dec 10 2009

No more terms up to k = 7500.

14286, 30617, and 37815, found by Daniel Heuer in 2001-2002, are also terms; see Caldwell link. - Jeppe Stig Nielsen, Jan 30 2022

Chris K. Caldwell, Prime Pages, Database Search Output

Cf. A171376, A171411, A171459, A171554.

Select[Range[1000], PrimeQ[1 + 8 10^# + 100^# ] &] (* Vincenzo Librandi, Apr 26 2014 *)

is(n)=ispseudoprime(1+8*10^n+100^n) \\ Charles R Greathouse IV, Jun 13 2017

a(11)-a(13) from Lucas A. Brown, Mar 04 2024

A171554 Numbers k such that 1 + 9*10^k + 100^k is prime.

Original entry on oeis.org

0, 1, 5, 71, 311

Offset: 1

Jason Earls, Dec 11 2009

No more terms up to k = 7200.

No more terms up to k = 27200. - Michael S. Branicky, Jun 30 2024

Cf. A171376, A171411, A171459, A171514.

Select[Range[0,320],PrimeQ[1+9 10^#+100^#]&] (* Harvey P. Dale, Nov 23 2023 *)

is(n)=ispseudoprime(1+9*10^n+100^n) \\ Charles R Greathouse IV, Jun 13 2017

A261455 Numbers n such that 1+131*10^(n-1)+100^n is prime.

Original entry on oeis.org

2, 4, 13, 30, 73, 82, 120, 168, 227, 422, 433, 451, 607, 612, 798, 1527, 6958

Offset: 1

Chai Wah Wu, Aug 20 2015

No other terms < 10000.

			For n=2, 11311 is prime.
For n=3, 1013101 is not prime.
For n=4, 100131001 is prime.

Cf. A171411, A171376.

isok(n)=isprime(1+131*10^(n-1)+100^n) \\ Anders Hellström, Aug 20 2015

cp's OEIS Frontend

A100028 Values of n for which the decimal number 10...030...01 is an n-digit prime.

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A070247 Palindromic primes with digit sum 5.

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A171411 Numbers k such that 1 + 5*10^k + 100^k is prime.

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A171459 Numbers k such that 1 + 6*10^k + 100^k is prime.

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A171514 Numbers k such that 1 + 8*10^k + 100^k is prime.

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A171554 Numbers k such that 1 + 9*10^k + 100^k is prime.

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A261455 Numbers n such that 1+131*10^(n-1)+100^n is prime.

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