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List of pairs of prime numbers (p,q) starting with (2, 3) such that p || q (where || denotes concatenation) is a prime number and the sequence is always extended with the smallest prime not yet present in the sequence.
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%I #70 Jul 03 2024 20:06:40

%S 2,3,5,23,7,19,11,17,13,61,29,53,31,37,41,59,43,73,47,83,67,79,71,167,

%T 89,101,97,103,107,137,109,139,113,131,127,157,149,173,151,163,179,

%U 233,181,193,191,227,197,257,199,211,223,229,239,251,241,271,263,269,277,331,281,317,283,397,293,311,307,337,313,373,347,359,349,379,353

%N List of pairs of prime numbers (p,q) starting with (2, 3) such that p || q (where || denotes concatenation) is a prime number and the sequence is always extended with the smallest prime not yet present in the sequence.

%H Paolo P. Lava, <a href="/A244862/b244862.txt">Table of n, a(n) for n = 1..1000</a>

%e The first few pairs are (2,3),(5,23),(7,19),(11,17),(13,61),(29,53), ..., giving the primes 23, 523, 719, 1117, 1361, 2953, ...

%p with(numtheory):nn:=60:lst:={2,3}: printf ( "%d %d \n",2,3):

%p for a from 2 to nn do:

%p p:=ithprime(a):ii:=0:

%p for b from 1 to nn while(ii=0)do:

%p q:=ithprime(b):s:=p*10^(length(q))+q:

%p if type(s,prime)=true and lst intersect {p,q}={}

%p then

%p lst:=lst union {p,q}:ii:=1:printf(`%d, `,p):printf(`%d, `,q):

%p else

%p fi:

%p od:

%p od:

%p [I have been informed that this program may be incorrect. - _N. J. A. Sloane_, Jul 03 2024]

%p # alternative version

%p P:=proc(q) local a,b,k,i,j,n,ok; a:=[2,3];

%p for n from 1 to q do k:=3; ok:=1; for i do if ok=1 then k:=nextprime(k);

%p if numboccur(k,a)=0 then b:=k;

%p for j from k do k:=nextprime(k); if numboccur(k,a)=0 then

%p if isprime(b*10^length(k)+k) then a:=[op(a),b,k]; ok:=0; break; fi; fi; od; fi;

%p else break;fi; od; od; print(op(a)); end: P(500); # _Paolo P. Lava_, Jul 03 2024

%Y Cf. A000040, A105184.

%Y A373794 is a very similar sequence (they first differ at term 69).

%K nonn,base,tabf

%O 1,1

%A _Michel Lagneau_, Jul 25 2014

%E Edited by _N. J. A. Sloane_, Jul 03 2024. More than the usual number of terms are shown in order to distinguish this from A373794.