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

2, 3, 11, 7, 41, 31, 17, 163, 23, 79, 197, 241, 29, 37, 59, 193, 227, 229, 239, 439, 929, 337, 257, 1447, 509, 19, 293, 1723, 1619, 937, 179, 367, 251, 1063, 4241, 1291, 521, 1951, 443, 139, 191, 1753, 1217, 673, 53, 883, 809, 109, 5381, 3733, 311, 967, 449

Offset: 1

Author

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 06 2003

Keywords

• base
• nonn

Comments

Conjecture: every prime except 5 is a term.

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

Examples

			2 is a prime.
2||3 = 23 is a prime.
2||3||7 = 3*79 but 2||3||11 = 2311 is a prime
So is 23117.  And so on.
		

Links

• Paul Zimmermann, Table of n, a(n) for n = 1..1479 [First 800 terms from Giorgos Kalogeropoulos; first 1156 terms from Metin Sariyar]

Crossrefs

Cf. A051670, A083759, A338072, A338073, A051670.

See also A158521.

Programs

Mathematica

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 *)

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))};
a083758(53) \\ Hugo Pfoertner, Oct 21 2020

Extensions

More terms from Sean A. Irvine, Dec 15 2009

Edited by N. J. A. Sloane, Oct 19 2020 following a comment from David James Sycamore