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

%I A374238 #99 Nov 03 2024 19:26:37
%S A374238 11,3333311,7771717,11818181,515115551,727722727,887887787,1110011101,
%T A374238 1161611161,1411111441,1411141411,1717117117,1911999919,3311113111,
%U A374238 3313133311,3333353533,5151111551,5555115151,5777777557,7373733337,7747447777,7777111777,8887788787,9199119991,9994449499
%N A374238 Primes whose pattern of identical digits is unique among the primes.
%C A374238 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.
%C A374238 Prime repunits (A004022) are a subsequence, as they are the sole primes with a single distinct digit.
%C A374238 A cryptarithm (alphametic) expresses a digit pattern in letters, where each distinct letter is to map to a distinct digit.
%C A374238 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.
%C A374238 The smallest term with 3 distinct digits is 1151135331533311.
%C A374238 The number of terms of length n is given by A376084(n).
%H A374238 Dmytro Inosov, <a href="/A374238/b374238.txt">Table of n, a(n) for n = 1..154</a>
%H A374238 Dmytro Inosov, <a href="/A374238/a374238.txt">Table of n, a(n) for n = 1..24840</a>
%H A374238 Dmytro S. Inosov and Emil Vlasák, <a href="https://arxiv.org/abs/2410.21427">Cryptarithmically unique terms in integer sequences</a>, arXiv:2410.21427 [math.NT], 2024.
%H A374238 Wikipedia, <a href="https://en.wikipedia.org/wiki/Verbal_arithmetic">Verbal arithmetic</a>.
%e A374238 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).
%e A374238 Counterexample: 13 is not a term since another prime 17 has the same pattern "AB" of two nonidentical digits.
%e A374238 7771717 is a term since it's prime and no other prime has the same pattern "AAABABA".
%t A374238 NumOfDigits = 10; (*Maximal integer length to be searched for*)
%t A374238 A358497[k_] :=
%t A374238   FromDigits[
%t A374238    Table[Mod[
%t A374238      CountDistinct[Take[#, FirstPosition[#, #[[i]]][[1]]]] &[
%t A374238       IntegerDigits[k]], 10], {i, 1, IntegerLength[k]}]];
%t A374238 A006880[MaxLen_] := PrimePi[10^MaxLen];
%t A374238 Extract[Select[
%t A374238    Tally[Table[{#, A358497[#]} &[Prime[i]], {i, 1,
%t A374238        A006880[NumOfDigits]}], #1[[2]] == #2[[2]] &], #[[2]] == 1 &], {All, 1}]
%Y A374238 Cf. A000040 (primes), A004022 (prime repunits), A358497, A039986, A376918, A376084, A376118.
%K A374238 nonn,base
%O A374238 1,1
%A A374238 _Dmytro Inosov_, Jul 01 2024