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A360534 Lexicographically earliest sequence of distinct prime numbers such that among each pair of consecutive terms, the decimal expansion of the smallest term appears in that of the largest term.

%I A360534 #10 Feb 12 2023 10:05:53
%S A360534 2,23,3,13,113,11,211,2111,22111,322111,3221,32213,2213,22133,622133,
%T A360534 6221,62213,362213,5362213,5,53,353,3533,33533,333533,33353,233353,
%U A360534 233,2333,23333,323333,3233333,32333333,632333333,6323,86323,863,3863,33863,1338637,7
%N A360534 Lexicographically earliest sequence of distinct prime numbers such that among each pair of consecutive terms, the decimal expansion of the smallest term appears in that of the largest term.
%C A360534 This sequence is infinite:
%C A360534 - if a(n) < 10^k, then 10^(k+1) and 10*a(n) + 1 are coprime,
%C A360534 - so, by Dirichlet's theorem on arithmetic progressions, there are infinitely many prime numbers of the form k*10^(k+1) + 10*a(n) + 1, and we can extend the sequence.
%C A360534 If we consider positive integers instead of prime numbers, then we obtain the powers of 10 (A011557).
%H A360534 Rémy Sigrist, <a href="/A360534/b360534.txt">Table of n, a(n) for n = 1..10000</a>
%H A360534 Rémy Sigrist, <a href="/A360534/a360534.gp.txt">PARI program</a>
%e A360534 The first terms are:
%e A360534   n   a(n)    a(n) aligned
%e A360534   --  ------  ------------
%e A360534    1       2        2
%e A360534    2      23        23
%e A360534    3       3         3
%e A360534    4      13        13
%e A360534    5     113       113
%e A360534    6      11       11
%e A360534    7     211      211
%e A360534    8    2111      2111
%e A360534    9   22111     22111
%e A360534   10  322111    322111
%e A360534   11    3221    3221
%e A360534   12   32213    32213
%e A360534   13    2213     2213
%e A360534   14   22133    22133
%e A360534   15  622133   622133
%o A360534 (PARI) See Links section.
%Y A360534 Cf. A011557, A030456, A335043.
%K A360534 nonn,base
%O A360534 1,1
%A A360534 _Rémy Sigrist_, Feb 10 2023