A246806 -id:A246806 - cp's OEIS frontend

A152242 Integers formed by concatenating primes.

2, 3, 5, 7, 11, 13, 17, 19, 22, 23, 25, 27, 29, 31, 32, 33, 35, 37, 41, 43, 47, 52, 53, 55, 57, 59, 61, 67, 71, 72, 73, 75, 77, 79, 83, 89, 97, 101, 103, 107, 109, 112, 113, 115, 117, 127, 131, 132, 133, 135, 137, 139, 149, 151, 157, 163, 167, 172, 173, 175, 177, 179

Offset: 1

Eric Angelini, Oct 15 2009

Leading zeros are not allowed (cf. A166504).

For any k > 0, there are A246806(k) terms with k digits. - Rémy Sigrist, Jan 08 2023

			101 is a member since it is prime; 303 is not since it is composite and 30 is also not a prime.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Henri Picciotto, Selected Integer Sequences

Cf. A019549, A038692, A066737, A105184, A129800, A166504, A246806.

is_A152242(n)=/* If n is even, the last digit must be 2 and [n\10] (if nonzero) must be in this sequence. (This check is not necessary but improves speed.) */ bittest(n,0) || return( n%10==2 && (n<10 || is_A152242(n\10))); isprime(n) && return(1); for(i=1,#Str(n)-1, n%10^i>10^(i-1) && isprime( n%10^i ) && is_A152242( n\10^i) && return(1)) \\ M. F. Hasler, Oct 15 2009; edited Oct 16 2009, to disallow leading zeros

from sympy import isprime
def ok(n):
    if isprime(n): return True
    s = str(n)
    return any(s[i]!="0" and isprime(int(s[:i])) and ok(int(s[i:])) for i in range(1, len(s)))
print([k for k in range(180) if ok(k)]) # Michael S. Branicky, Sep 01 2024

More terms from M. F. Hasler and Zak Seidov, Oct 15 2009

A246807 Number of n-bit numbers that can be written as the concatenation of 0 or more prime numbers (everything written in base 2).

Original entry on oeis.org

1, 0, 2, 2, 5, 8, 15, 33, 59, 126, 246, 494, 978, 1971, 3930, 7845, 15749, 31527, 63349, 126986, 254880, 511468, 1026348, 2060633, 4135808, 8303940, 16669925, 33472231, 67201664, 134930088, 270895845, 543915707, 1091923726, 2192302476, 4400938402, 8835035284

Offset: 0

Jeffrey Shallit, Nov 16 2014

Here we only consider canonical base-2 expansions (with no leading zeros). 1 is not a prime, and neither is 0.

			For n = 5 the 8 solutions counted include the primes {17,19,23,29,31} between 16 and 31, and also the numbers 21 (10.101), 22 (101.10), and 30 (111.10).

Cf. A162145, A246806.

from sympy import isprime, primerange
from functools import lru_cache
@lru_cache(maxsize=None)
def ok(n):
  if n%4 == 0: return False
  if isprime(n): return True
  b = bin(n)[2:]
  for i in range(2, len(b)-1):
    if b[i] != '0' and isprime(int(b[:i], 2)) and ok(int(b[i:], 2)):
      return True
  return False
def a(n):
  return 1 if n == 0 else sum(1 for m in range(2**(n-1), 2**n) if ok(m))
print([a(n) for n in range(21)]) # Michael S. Branicky, Mar 26 2021

More terms from Jeffrey Shallit, Nov 25 2014
a(29)-a(32) from Michael S. Branicky, Mar 26 2021
a(33)-a(35) from Jinyuan Wang, May 31 2025

cp's OEIS Frontend

A152242 Integers formed by concatenating primes.

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