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A348358 Primes which are not the concatenation of smaller primes (in base 10 and allowing leading 0's).

%I A348358 #23 Oct 16 2021 03:32:16
%S A348358 2,3,5,7,11,13,17,19,29,31,41,43,47,59,61,67,71,79,83,89,97,101,103,
%T A348358 107,109,127,131,139,149,151,157,163,167,179,181,191,199,239,251,263,
%U A348358 269,281,349,401,409,419,421,431,439,443,449,457,461,463,467,479,487,491,499
%N A348358 Primes which are not the concatenation of smaller primes (in base 10 and allowing leading 0's).
%C A348358 This is the sequence of numbers that are neither a product of smaller primes nor a concatenation of smaller primes (in base 10).
%C A348358 This sequence differs from A238647. The prime 227 is in A238647 but not in this sequence for it is the concatenation of primes 2, 2, 7 (in base 10).
%C A348358 Conjecture. If p > 7 is a prime, then there exists a base b such that p in base b is the concatenation of smaller primes in base b.
%e A348358 The prime 127 is in the sequence because the only expressions of 127 as concatenation of smaller numbers are 1 U 2 U 7, 1 U 27, and 12 U 7 (in base 10) but 1 and 12 are not primes.
%e A348358 The prime 271 is not in the sequence because it is the concatenation of primes 2 and 71 (in base 10).
%e A348358 The prime 307 is not in the sequence because it is the concatenation of primes 3 and 07 (in base 10).
%t A348358 Select[Prime@Range@100,Union[And@@@PrimeQ[FromDigits/@#&/@Union@Select[Flatten[Permutations/@Subsets[Most@Rest@Subsequences[d=IntegerDigits@#]],1],Flatten@#==d&]]]=={False}||Length@d==1&] (* _Giorgos Kalogeropoulos_, Oct 15 2021 *)
%o A348358 (Python)
%o A348358 from sympy import isprime, primerange
%o A348358 def cond(n): # n is not a concatenation of smaller primes
%o A348358     if n%10 in {4, 6, 8}: return True
%o A348358     d = str(n)
%o A348358     for i in range(1, len(d)):
%o A348358         if isprime(int(d[:i])):
%o A348358              if isprime(int(d[i:])) or not cond(int(d[i:])):
%o A348358                  return False
%o A348358     return True
%o A348358 def aupto(lim): return [p for p in primerange(2, lim+1) if cond(p)]
%o A348358 print(aupto(490)) # _Michael S. Branicky_, Oct 15 2021
%Y A348358 Cf. A141033, A141409, A238647, A342244, A348358.
%K A348358 easy,nonn,base
%O A348358 1,1
%A A348358 _M. Farrokhi D. G._, Oct 14 2021