A100028 -id:A100028 - cp's OEIS frontend

A100026 Consider all (2n+1)-digit palindromic primes of the form 10...0M0...01 (so that M is a palindrome with <= 2n-1 digits); a(n) = smallest such M.

0, 3, 3, 3, 5, 8, 323, 5, 8, 212, 3, 161, 8, 3, 242, 3, 8, 10901, 737, 161, 242, 333, 282, 6, 252, 474, 5, 12921, 8, 131, 18381, 6, 444, 6, 797, 606, 717, 15351, 464, 333, 626, 545, 13031, 161, 747, 191, 323, 636, 32523, 303, 282, 888, 686, 18981, 111, 15951, 12021

Offset: 1

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

Is this the same as "Longest palindromic proper substring of A100027(n) or A028989(n+1) that occurs only once in the decimal representation of A100027(n) or A028989(n+1), respectively"? - Felix Fröhlich, Apr 30 2022

A more formal definition may be a(n) = A004151(A028989(n+1) - 10^(2n) - 1) with the convention that A004151(0) = 0. Only in the unlikely situation that A080176 contains undiscovered primes will a(n) = 0 occur for n > 1. - Jeppe Stig Nielsen, Apr 04 2025

Chai Wah Wu, Table of n, a(n) for n = 1..1000

The corresponding palindromic primes are shown in A100027.

Cf. A028989, A099744, A099746, A100028.

f[n_] := Block[{k = 0, t = Flatten[Join[{1}, Table[0, {n - 1}]]]}, While[s = Drop[t, Min[ -Floor[ Log[10, k]/2], 0]]; k != FromDigits[ Reverse[ IntegerDigits[k]]] || !PrimeQ[ FromDigits[ Join[s, IntegerDigits[k], Reverse[s]]]], k++ ]; k]; Table[ f[n], {n, 56}] (* Robert G. Wilson v, Nov 22 2004 *)

More terms from Robert G. Wilson v, Nov 22 2004

A100027 Smallest (2n+1)-digit palindromic prime of the form 10...0M0...01 (thus M is a palindrome with <= 2n-1 digits).

Original entry on oeis.org

101, 10301, 1003001, 100030001, 10000500001, 1000008000001, 100000323000001, 10000000500000001, 1000000008000000001, 100000000212000000001, 10000000000300000000001, 1000000000016100000000001, 100000000000080000000000001, 10000000000000300000000000001, 1000000000000024200000000000001

Offset: 1

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

Essentially a duplicate of A028989. - Jeppe Stig Nielsen, Apr 04 2025

Values of M are given in A100026. Cf. A100028.

Cf. A028989.

A171376 Numbers k such that 1 + 3*10^k + 100^k is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 11, 14, 16, 92, 133, 153, 378, 448, 785, 1488, 1915, 2297, 3286, 4755, 5825, 7820, 34442, 34941

Offset: 1

Jason Earls, Dec 07 2009

All primes were certified with WinPFGW.

a(24) > 35000. - Serge Batalov, Dec 20 2015

			4 is in the sequence because 10^8 + 3 * 10^4 + 1 = 100030001 is prime.

Chris K. Caldwell, Prime Pages, Database Search Output

Cf. A082622.

Cf. A171411, A171459, A171514, A171554.

[n: n in [0..4*10^2] | IsPrime(1+3*10^n+100^n)]; // Vincenzo Librandi, Dec 22 2015

Select[Range@ 1000, PrimeQ[1 + 3 10^# + 100^#] &] (* Michael De Vlieger, Dec 18 2015 *)

\\sieve for the candidates:
{
lim=10^9; ns=6*10^5; pp=10^7; s=vectorsmall(ns);
forprime(p=11,lim,if(kronecker(5,p)==1,o=znorder(t=Mod(10,p));
  q=sqrt(Mod(5,p));r=znlog((q-3)/2,t,o);
  if(r,forstep(n=r,ns,o,s[n]=1);forstep(n=o-r,ns,o,s[n]=1)));
  if(p>pp,pp+=10000000;print1(p" ")));
for(n=1,ns,if(!s[n],write("sieve_out_10301NGm1.txt", n)));
}
\\quick initial check for small sequence members
for(n=0,2297,if(ispseudoprime((10^n+3)*10^n+1),print1(n", ")))
\\ Serge Batalov, Dec 17 2015

a(n) = (A100028(n-1) - 1)/2 for n>1. - Jeppe Stig Nielsen, Oct 06 2024

a(21)-a(23) from Serge Batalov, Dec 20 2015

A100459 Values of n for which the decimal number 10...090...01 is an n-digit prime.

Original entry on oeis.org

3, 11, 143, 623

Offset: 1

Harvey Dubner (harvey(AT)dubner.com), Nov 23 2004

a(5) > 84309. - Robert Price, Apr 14 2013

The number n must be of the form 2x+3, by definition. If x is odd, the number 10...090...01 is divisible by 11. Also, if x == 2 mod 6, it is divisible by 7. Thus, it must be either 4 mod 6 or 0 mod 6. Thus, 2*(6x+4) + 3 = 12x + 11 and 2*(6x+0) + 3 = 12x + 3, implying that a(n) must be congruent to 11 mod 12 or 3 mod 12. - Derek Orr, Jul 28 2014

			10000900001 is an 11-digit prime. Thus 11 is a member of this sequence. - _Derek Orr_, Jul 27 2014

Cf. A100028, A171553.

2Select[Range[1000], PrimeQ[100^# + 9 * 10^# + 1] &] + 1 (* Alonso del Arte, Jul 28 2014 *)

for(n=0,10^5,p=10^(2*n+2)+9*10^(n+1)+1;if(ispseudoprime(p),print1(2*n+3,", "))) \\ Derek Orr, Jul 27 2014

A100456 Values of n for which the decimal number 10...050...01 is an n-digit prime.

Original entry on oeis.org

3, 5, 9, 11, 17, 55, 331, 461, 475, 739, 971, 1257, 1751, 1929, 2839, 8167

Offset: 1

Harvey Dubner (harvey(AT)dubner.com), Nov 23 2004

a(17) > 53125. - Robert Price, Apr 14 2013

Three is a term because 151 is a prime, thus illustrating that zeros are optional in the resulting primes. - Harvey P. Dale, Oct 08 2019

Cf. A100028.

IntegerLength/@Select[Table[FromDigits[Join[PadRight[{1},n,0],{5},PadLeft[ {1},n,0]]],{n,5000}],PrimeQ] (* Harvey P. Dale, Oct 08 2019 *)

Corrected by T. D. Noe, Oct 25 2006

A100458 Values of n for which the decimal number 10...080...01 is an n-digit prime.

Original entry on oeis.org

3, 7, 13, 19, 27, 35, 59, 127, 181, 1063, 28573

Offset: 1

Harvey Dubner (harvey(AT)dubner.com), Nov 23 2004

a(12) > 54647. - Robert Price, Apr 14 2013

Cf. A100028.

A100457 Values of n for which the decimal number 10...060...01 is an n-digit prime.

Original entry on oeis.org

5, 9, 49, 65, 69, 145, 151, 329, 1065, 2671, 9409

Offset: 1

Harvey Dubner (harvey(AT)dubner.com), Nov 23 2004

a(12) > 54299. - Robert Price, Apr 14 2013

Cf. A100028.

a(11) from Robert Price, Apr 14 2013

A261450 Smallest k such that A011557(n)//k//rev is prime, where rev is the string of digits of A011557(n) reversed (retaining any leading zeros) and // denotes concatenation.

Original entry on oeis.org

0, 3, 3, 3, 5, 8, 29, 5, 8, 15, 3, 21, 8, 3, 21, 3, 8, 18, 20, 92, 110, 51, 102, 6, 57, 23, 5, 114, 8, 32, 41, 6, 236, 6, 39, 60, 110, 62, 36, 17, 53, 21, 161, 41, 159, 57, 137, 42, 83, 114, 126, 80, 30, 36, 278, 107, 425, 111, 68, 68, 95, 29, 8, 53, 426, 48

Offset: 0

Felix Fröhlich, Aug 23 2015

Is a(n) = 0 for any n > 0? If such an n exists, that n is a term of A000079 (cf. Greathouse, 2010).
All terms are congruent to 0 or 2 modulo 3, since if k is congruent to 1 modulo 3, 1000...0//k//00...01 is divisible by 3 and thus not prime.
a(n) <= A100026(n-1) with equality when a(n) is a palindrome. - Michel Marcus, Sep 11 2015

			a(1) = 3, because 10001, 10101, and 10201 are composite and 10301 is prime.
a(6) = 29, because 29 is the smallest k such that 1000000//k//0000001 is prime. The decimal expansion of that prime is 1000000290000001.

Dana Jacobsen, Table of n, a(n) for n = 0..500
C. R. Greathouse IV, Primes of form 10^k + 1?, Physics Forums (message from Apr 6, 2010).

Cf. A000079, A011557, A100026, A100028, A100456, A100458, A100459, A258372.

Table[k = 0; d = IntegerDigits[10^n]; While[! PrimeQ@ FromDigits@ Join[d, IntegerDigits@ k, Reverse@ d], k++]; k, {n, 0, 65}] (* Michael De Vlieger, Aug 26 2015 *)

a(n) = x=10^n; k=0; while(!ispseudoprime(eval(Str(x, k, concat(Vecrev(Str(x)))))), k++); k

use ntheory ":all"; for my $n (0..50) { my($t,$c)=(0); $t++ while $c=1 . 0 x $n . $t . 0 x $n . 1, !is_prob_prime($c); say "$n $t"; } # Dana Jacobsen, Oct 02 2015

cp's OEIS Frontend

A100026 Consider all (2n+1)-digit palindromic primes of the form 10...0M0...01 (so that M is a palindrome with <= 2n-1 digits); a(n) = smallest such M.

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A100027 Smallest (2n+1)-digit palindromic prime of the form 10...0M0...01 (thus M is a palindrome with <= 2n-1 digits).

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A171376 Numbers k such that 1 + 3*10^k + 100^k is prime.

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A100459 Values of n for which the decimal number 10...090...01 is an n-digit prime.

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A100456 Values of n for which the decimal number 10...050...01 is an n-digit prime.

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A100458 Values of n for which the decimal number 10...080...01 is an n-digit prime.

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A100457 Values of n for which the decimal number 10...060...01 is an n-digit prime.

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Author

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Comments

Crossrefs

Extensions

A261450 Smallest k such that A011557(n)//k//rev is prime, where rev is the string of digits of A011557(n) reversed (retaining any leading zeros) and // denotes concatenation.

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