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A385481 Primes whose decimal expansion consists of the concatenation of ij, iijj, iiijjj,..., and m i’s followed by m j’s, i != j, where 1<= i, j <= 9 and m > 0.

%I A385481 #20 Jul 05 2025 16:41:59
%S A385481 13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,
%T A385481 31331133311133331111,37337733377733337777,43443344433344443333,
%U A385481 49449944499944449999
%N A385481 Primes whose decimal expansion consists of the concatenation of ij, iijj, iiijjj,..., and m i’s followed by m j’s, i != j, where 1<= i, j <= 9 and m > 0.
%C A385481 a(25) has 812 digits and starts with 292299222999...999, where the last concatenated strings have 28 2's followed by 28 9's.
%C A385481 Since each term is prime, then j = 1, 3, 7, and 9 and i + j == 1, 2 (mod 3).
%C A385481 m == 1 (mod 3). Proof. If k is a term whose last concatenated string has m i's followed by m j's, then it has 2(1 + 2 + ... + m) = m(m + 1) digits, whose sum is (i + j)*m(m + 1)/2, so that if m == 0, 2 (mod 3), then the sum of digits of k is a multiple of 3 and so k is not prime.
%C A385481 Are there infinite primes of this form?
%C A385481 From _Michael S. Branicky_, Jul 01 2025: (Start)
%C A385481 A probabilistic argument suggests the sequence is finite.
%C A385481 a(26), if it exists, has m > 321 and > 103362 digits. (End)
%H A385481 Gonzalo Martínez, <a href="/A385481/b385481.txt">Table of n, a(n) for n = 1..25</a>
%e A385481 for i = 3, j = 1 and m = 4, by concatenating 31, 3311, 333111, 33331111 the prime 31331133311133331111 is obtained.
%o A385481 (Python)
%o A385481 from gmpy2 import is_prime, mpz
%o A385481 from itertools import count, islice, product
%o A385481 def agen(): yield from (p for m in count(1) for i in "123456789" for j in "1379" if i != j and is_prime(p:=int(mpz("".join(i*k+j*k for k in range(1, m+1))))))
%o A385481 print(list(islice(agen(), 24))) # _Michael S. Branicky_, Jun 30 2025
%Y A385481 Cf. A059170, A034845.
%K A385481 nonn,base
%O A385481 1,1
%A A385481 _Gonzalo Martínez_, Jun 30 2025