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A083754 a(1) = 1 and then smallest odd number not occurring earlier such that the concatenation a(1)a(2)a(3)... is a prime.

%I A083754 #13 Oct 30 2020 22:57:56
%S A083754 1,3,7,11,9,27,63,31,53,21,13,83,33,39,49,51,77,87,307,29,229,281,151,
%T A083754 173,481,41,99,157,177,17,357,213,231,171,271,557,67,113,463,159,119,
%U A083754 57,247,147,563,409,353,391,179,1051,209,19,153,621,287,567,313,117,363
%N A083754 a(1) = 1 and then smallest odd number not occurring earlier such that the concatenation a(1)a(2)a(3)... is a prime.
%C A083754 Conjecture: all odd numbers not of the type 10k+5 are members.
%C A083754 Some of the larger entries may only correspond to probable primes.
%C A083754 Values corresponding to a(6)=27 (A083755(5)) through a(59)=363 (A083755(58), a 149-digit value) have been certified prime with Primo. - _Rick L. Shepherd_, May 10 2003
%C A083754 Since we begin with 1 and thereafter have more than a single decimal digit, all terms must be in A045572, the sequence that contains all positive integers relatively prime to 10. - _Michael De Vlieger_, Oct 30 2020.
%H A083754 Michael De Vlieger, <a href="/A083754/b083754.txt">Table of n, a(n) for n = 1..501</a>
%H A083754 Michael De Vlieger, <a href="/A083754/a083754.png">Plot of a(n) for 1 <= n <= 500</a>, showing congruency of a(n) with 1 (orange), 3 (green), 7 (blue), or 9 (magenta) (mod 10).
%e A083754 13,137,13711, etc. are primes.(1379 is not a prime) hence 11 is the next member after 7.
%t A083754 Block[{c = 1, a = {1}, f, g}, f[m_, n_] := m*10^(1 + Floor[Log10[n]]) + n; g[n_] := (5 n + Mod[3 n + 2, 4] - 4)/2; Do[Block[{j = 2, k, d, t}, While[Nand[FreeQ[a, Set[k, g[j] ]], PrimeQ[Set[d, f[c, k]]]], j++]; c = d; AppendTo[a, k]], {i, 59}]; a] (* _Michael De Vlieger_, Oct 30 2020 *)
%o A083754 (PARI) {used_before(v, n) = for (l=1,matsize(v)[2], if(v[l]==n, return(1))); return(0)} {A083754=[1]; p=A083754[1]; A083755=[]; print1(A083754[1],","); for (m=2,151, k=1; while (used_before(A083754,k)||!isprime(tmp_p=p*(10^length(Str(k)))+k), k=k+2); p=tmp_p; A083755=concat(A083755,p); A083754=concat(A083754,k); print1(A083754[m],",")); A083755}
%Y A083754 Cf. A045572, A083755.
%K A083754 base,nonn
%O A083754 1,2
%A A083754 _Amarnath Murthy_ and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 06 2003
%E A083754 More terms from _Rick L. Shepherd_, May 08 2003