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A195302 Xmas tree primes.

%I A195302 #38 Jun 05 2025 13:46:25
%S A195302 2,3,5,7,211,223,229,241,271,283,311,313,317,331,337,347,353,359,367,
%T A195302 373,379,383,389,397,523,541,547,571,719,743,761,773,797,211151,
%U A195302 211193,211199,211229,211241,211271,211283,211313,211349,211373,211433,211457,211499,211571,211619,211643,211661,211691,211727,211811,211859,211877,211997,213131
%N A195302 Xmas tree primes.
%C A195302 A Xmas tree prime is a prime which is a concatenation of a prime with a single digit, a prime with two digits, a prime with three digits, a prime with four digits etc. By definition, the number of digits is a triangular number (A000217). Leading zeros are not allowed for any of the primes.
%H A195302 Alvin Hoover Belt, <a href="/A195302/b195302.txt">Table of n, a(n) for n = 1..100</a>
%H A195302 Terry Trotter, <a href="http://trottermath.net/numtrivia/potpourri.html">POTPOURRI</a>
%e A195302 359 is a Xmas tree prime because it is prime and 3 and 59 are prime.
%e A195302 503 is not a Xmas tree prime although 5 and 3 are prime, because the leading 0 in front of the 3 is not allowed by definition.
%p A195302 isA000217 := proc(n)
%p A195302         for k from 0 do
%p A195302                 if n = k*(k+1)/2 then
%p A195302                         return k;
%p A195302                 elif n < k*(k+1)/2 then
%p A195302                         return -1 ;
%p A195302                 end if;
%p A195302         end do;
%p A195302 end proc:
%p A195302 isA195302 := proc(n)
%p A195302         local dgs,T,d,std,kList,k ;
%p A195302         if isprime(n) then
%p A195302                 dgs := convert(n,base,10) ;
%p A195302                 T := isA000217(nops(dgs)) ;
%p A195302                 if T > 0 then
%p A195302                         std := 1 ;
%p A195302                         for d from T to 1 by -1 do
%p A195302                                 kList := [op(std..std+d-1,dgs)] ;
%p A195302                                 if op(-1,kList) = 0 then
%p A195302                                         return false;
%p A195302                                 end if;
%p A195302                                 k := add(op(i,kList)*10^(i-1),i=1..nops(kList)) ;
%p A195302                                 if not isprime(k) then
%p A195302                                         return false;
%p A195302                                 end if;
%p A195302                                 std := std+d ;
%p A195302                         end do:
%p A195302                         return true;
%p A195302                 else
%p A195302                         false;
%p A195302                 end if;
%p A195302         else
%p A195302                 false;
%p A195302         end if;
%p A195302 end proc:
%p A195302 for i from 2 to 300000 do
%p A195302         if isA195302(i) then
%p A195302                 printf("%d,",i) ;
%p A195302         end if;
%p A195302 end do: # _R. J. Mathar_, Sep 20 2011
%o A195302 (Python)
%o A195302 from sympy import isprime, sieve
%o A195302 from itertools import product
%o A195302 def alst(n):
%o A195302   alst, plen = [], 1
%o A195302   while True:
%o A195302     sieve.extend(10**plen-1)
%o A195302     primes = list(str(p) for p in sieve._list)
%o A195302     primesbylen = [[p for p in primes if len(p)==i+1] for i in range(plen)]
%o A195302     for t in product(*primesbylen):
%o A195302       intt = int("".join(t))
%o A195302       if isprime(intt): alst.append(intt)
%o A195302       if len(alst) == n: return alst
%o A195302     plen += 1
%o A195302 print(alst(57)) # _Michael S. Branicky_, Dec 26 2020
%Y A195302 Cf. A195335.
%K A195302 nonn,base
%O A195302 1,1
%A A195302 _Kausthub Gudipati_, Sep 16 2011