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A083758 Lexicographically earliest infinite sequence of distinct primes such that the concatenation of the initial n terms is a prime for all n >= 1.

%I A083758 #76 Aug 24 2024 01:54:05
%S A083758 2,3,11,7,41,31,17,163,23,79,197,241,29,37,59,193,227,229,239,439,929,
%T A083758 337,257,1447,509,19,293,1723,1619,937,179,367,251,1063,4241,1291,521,
%U A083758 1951,443,139,191,1753,1217,673,53,883,809,109,5381,3733,311,967,449
%N A083758 Lexicographically earliest infinite sequence of distinct primes such that the concatenation of the initial n terms is a prime for all n >= 1.
%C A083758 Conjecture: every prime except 5 is a term.
%C A083758 However, after 1000 terms, 13, 47, 61, ... are still missing. A158521 suggests there is no intrinsic reason why 13 should not eventually appear. - _N. J. A. Sloane_, Oct 21 2020
%H A083758 Paul Zimmermann, <a href="/A083758/b083758.txt">Table of n, a(n) for n = 1..1479</a> [First 800 terms from Giorgos Kalogeropoulos; first 1156 terms from Metin Sariyar]
%e A083758 2 is a prime.
%e A083758 2||3 = 23 is a prime.
%e A083758 2||3||7 = 3*79 but 2||3||11 = 2311 is a prime
%e A083758 So is 23117.  And so on.
%t A083758 a[1]=2;a[n_]:=a[n]=Module[{v=1,k=Table[a[m],{m,n-1}]},While[PrimeQ[FromDigits@Join[Flatten[IntegerDigits/@k],IntegerDigits[t=Prime[v]]]]==False||MemberQ[k,t],v++];k=Join[k,{t}];t];Table[a[i],{i,60}]  (* _Giorgos Kalogeropoulos_, May 28 2019 *)
%o A083758 (PARI) a083758(m)={my(np=1000*m,pused=vectorsmall(np),digp=[]); for(n=1,m,my(found=0);for(k=1,np, if(!pused[k],my(add=digits(prime(k)),pc=concat(digp,add));if(ispseudoprime(fromdigits(pc)),print1(prime(k),", ");digp=pc;pused[k]=1;found=1;break)));if(!found,break))};
%o A083758 a083758(53) \\ _Hugo Pfoertner_, Oct 21 2020
%Y A083758 Cf. A051670, A083759, A338072, A338073, A051670.
%Y A083758 See also A158521.
%K A083758 base,nonn
%O A083758 1,1
%A A083758 _Amarnath Murthy_ and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 06 2003
%E A083758 More terms from _Sean A. Irvine_, Dec 15 2009
%E A083758 Edited by _N. J. A. Sloane_, Oct 19 2020 following a comment from _David James Sycamore_