A374238 Primes whose pattern of identical digits is unique among the primes.

11, 3333311, 7771717, 11818181, 515115551, 727722727, 887887787, 1110011101, 1161611161, 1411111441, 1411141411, 1717117117, 1911999919, 3311113111, 3313133311, 3333353533, 5151111551, 5555115151, 5777777557, 7373733337, 7747447777, 7777111777, 8887788787, 9199119991, 9994449499

Offset: 1

Author

Dmytro Inosov, Jul 01 2024

Keywords

• nonn
• base

Comments

The digit pattern for any natural number n is uniquely defined by the canonical form A358497(n), which enumerates digits according to their position of first occurrence. Each prime in this sequence has a unique digit pattern in the sense that no other prime has the same pattern.

Prime repunits (A004022) are a subsequence, as they are the sole primes with a single distinct digit.

A cryptarithm (alphametic) expresses a digit pattern in letters, where each distinct letter is to map to a distinct digit.

If a cryptarithmetic problem calls for a prime number, then the primes in this sequence are unique solutions, so we call these primes cryptarithmically unique.

The smallest term with 3 distinct digits is 1151135331533311.

The number of terms of length n is given by A376084(n).

Examples

			11 is a term since no other prime has the same pattern "AA" of two identical digits (any other AA is divisible by A > 1, hence nonprime).
Counterexample: 13 is not a term since another prime 17 has the same pattern "AB" of two nonidentical digits.
7771717 is a term since it's prime and no other prime has the same pattern "AAABABA".
		

Links

• Dmytro Inosov, Table of n, a(n) for n = 1..154
• Dmytro Inosov, Table of n, a(n) for n = 1..24840
• Dmytro S. Inosov and Emil Vlasák, Cryptarithmically unique terms in integer sequences, arXiv:2410.21427 [math.NT], 2024.
• Wikipedia, Verbal arithmetic.

Crossrefs

Cf. A000040 (primes), A004022 (prime repunits), A358497, A039986, A376918, A376084, A376118.

Programs

Mathematica

NumOfDigits = 10; (*Maximal integer length to be searched for*)
A358497[k_] :=
  FromDigits[
   Table[Mod[
     CountDistinct[Take[#, FirstPosition[#, #[[i]]][[1]]]] &[
      IntegerDigits[k]], 10], {i, 1, IntegerLength[k]}]];
A006880[MaxLen_] := PrimePi[10^MaxLen];
Extract[Select[
   Tally[Table[{#, A358497[#]} &[Prime[i]], {i, 1,
       A006880[NumOfDigits]}], #1[[2]] == #2[[2]] &], #[[2]] == 1 &], {All, 1}]