A348358 - OEIS

A348358

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

2, 3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 47, 59, 61, 67, 71, 79, 83, 89, 97, 101, 103, 107, 109, 127, 131, 139, 149, 151, 157, 163, 167, 179, 181, 191, 199, 239, 251, 263, 269, 281, 349, 401, 409, 419, 421, 431, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499

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OFFSET

1,1

COMMENTS

This is the sequence of numbers that are neither a product of smaller primes nor a concatenation of smaller primes (in base 10).

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).

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.

LINKS

Table of n, a(n) for n=1..59.

EXAMPLE

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.

The prime 271 is not in the sequence because it is the concatenation of primes 2 and 71 (in base 10).

The prime 307 is not in the sequence because it is the concatenation of primes 3 and 07 (in base 10).

MATHEMATICA

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 *)

PROG

(Python)

from sympy import isprime, primerange

def cond(n): # n is not a concatenation of smaller primes

if n%10 in {4, 6, 8}: return True

d = str(n)

for i in range(1, len(d)):

if isprime(int(d[:i])):

if isprime(int(d[i:])) or not cond(int(d[i:])):

return False

return True

def aupto(lim): return [p for p in primerange(2, lim+1) if cond(p)]