A376118 Cryptarithmically unique palindromic primes.

11, 11141414111, 11999199911, 13111311131, 1110110110111, 1141411141411, 1611116111161, 3113113113113, 3222223222223, 3533355533353, 7444477744447, 7767777777677, 7887787877887, 7999979799997, 9494994994949, 9779999999779, 118818181818811, 131133131331131, 944499494994449, 10000010101000001

Offset: 1

Author

Dmytro Inosov, Sep 11 2024

Keywords

• nonn
• base

Comments

Each prime in this sequence is simultaneously a palindrome in base 10 and has a unique decimal digit pattern A358497(a(n)) in the sense that no other prime has the same pattern.

All terms except 11 have an odd number of digits (cf. A002385).

Number of terms < 100^k: 1, 1, 1, 1, 1, 4, 16, 19, (92), (249), (416), (1093)... . The numbers in brackets are conjectured based on the calculated terms with 1, 2, or 3 distinct digits and the vanishing combinatorial probability of terms with 4 or more distinct digits at these lengths.

The smallest term with 3 distinct digits is 11155511521212511555111.

Examples

			11141414111 is a term since it's a palindromic prime and no other prime has the same pattern "AAABABABAAA" of repeating digits.
Counterexample: the palindromic prime 131 is not a term since another prime 151 has the same pattern "ABA" of repeating digits.
		

Links

• Dmytro Inosov, Table of n, a(n) for n = 1..20
• Dmytro Inosov, Table of n, a(n) for n = 1..1093

Crossrefs

Intersection of A374238 and A002113.

Subsequence of A002385.

Supersequence of A004022 (prime repunits).

Cf. A358497.

Programs

Mathematica

NumOfDigits = 13; (* Maximal integer length to be searched for *)
(* A function that calculates the canonical form A358497[n] *)
A358497[k_] := FromDigits@a358497C[k]
a358497C = Compile[{{k, _Integer}}, Module[{firstpos = ConstantArray[0, 10], digits = IntegerDigits[k], indx = 0}, Table[If[firstpos[[digits[[j]] + 1]] == 0, firstpos[[digits[[j]] + 1]] = Mod[++indx, 10]]; firstpos[[digits[[j]] + 1]], {j, Length[digits]}]]];
(* Extracting cryptarithmically uniqie terms from palindromic primes *)
UniquePalPrimes = {11};
Do[PalindromicPrimes = {}; CryptUnique = {};
  Do[If[PrimeQ[#], AppendTo[PalindromicPrimes,{#, A358497[#]}]]&[n*10^(IntegerLength[n]-1) +
     FromDigits@Rest@Reverse@IntegerDigits[n]], {n, 10^(k-1), 10^k-1}];
  CryptUnique = Extract[Extract[Select[Tally[PalindromicPrimes, #1[[2]]==#2[[2]] &], #[[2]]==1 &], {All,1}], {All,1}];
  UniquePalPrimes = Join[UniquePalPrimes, CryptUnique];
  (* Prints the number of results and the list of results for every integer length *)
  Print[{2k-1, Length[CryptUnique], CryptUnique}], {k, 2, (NumOfDigits+1)/2}];
UniquePalPrimes