A195302 Xmas tree primes.

2, 3, 5, 7, 211, 223, 229, 241, 271, 283, 311, 313, 317, 331, 337, 347, 353, 359, 367, 373, 379, 383, 389, 397, 523, 541, 547, 571, 719, 743, 761, 773, 797, 211151, 211193, 211199, 211229, 211241, 211271, 211283, 211313, 211349, 211373, 211433, 211457, 211499, 211571, 211619, 211643, 211661, 211691, 211727, 211811, 211859, 211877, 211997, 213131

Offset: 1

Author

Kausthub Gudipati, Sep 16 2011

Keywords

• nonn
• base

Comments

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.

Examples

			359 is a Xmas tree prime because it is prime and 3 and 59 are prime.
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.
		

Links

• Alvin Hoover Belt, Table of n, a(n) for n = 1..100
• Terry Trotter, POTPOURRI

Crossrefs

Cf. A195335.

Programs

Maple

isA000217 := proc(n)
        for k from 0 do
                if n = k*(k+1)/2 then
                        return k;
                elif n < k*(k+1)/2 then
                        return -1 ;
                end if;
        end do;
end proc:
isA195302 := proc(n)
        local dgs,T,d,std,kList,k ;
        if isprime(n) then
                dgs := convert(n,base,10) ;
                T := isA000217(nops(dgs)) ;
                if T > 0 then
                        std := 1 ;
                        for d from T to 1 by -1 do
                                kList := [op(std..std+d-1,dgs)] ;
                                if op(-1,kList) = 0 then
                                        return false;
                                end if;
                                k := add(op(i,kList)*10^(i-1),i=1..nops(kList)) ;
                                if not isprime(k) then
                                        return false;
                                end if;
                                std := std+d ;
                        end do:
                        return true;
                else
                        false;
                end if;
        else
                false;
        end if;
end proc:
for i from 2 to 300000 do
        if isA195302(i) then
                printf("%d,",i) ;
        end if;
end do: # R. J. Mathar, Sep 20 2011

Python

from sympy import isprime, sieve
from itertools import product
def alst(n):
  alst, plen = [], 1
  while True:
    sieve.extend(10**plen-1)
    primes = list(str(p) for p in sieve._list)
    primesbylen = [[p for p in primes if len(p)==i+1] for i in range(plen)]
    for t in product(*primesbylen):
      intt = int("".join(t))
      if isprime(intt): alst.append(intt)
      if len(alst) == n: return alst
    plen += 1
print(alst(57)) # Michael S. Branicky, Dec 26 2020