A090423 Primes that can be written in binary representation as concatenation of other primes.

11, 23, 29, 31, 43, 47, 59, 61, 71, 79, 83, 109, 113, 127, 151, 157, 167, 173, 179, 181, 191, 223, 229, 233, 239, 241, 251, 271, 283, 317, 337, 347, 349, 353, 359, 367, 373, 379, 383, 431, 433, 439, 457, 463, 467, 479, 487, 491, 499, 503, 509, 541, 563, 599, 607

Offset: 1

Author

Reinhard Zumkeller, Nov 30 2003

Keywords

• nonn
• base

Comments

A090418(a(n)) > 1; subsequence of A090421.

Examples

			337 is 101010001 in binary,
10 is 2,
10 is 2,
10001 is 17, partition is 10_10_10001, so 337 is in the sequence.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A090422, A000040, A004676, A007088.

Programs

Haskell

a090423 n = a090423_list !! (n-1)
a090423_list = filter ((> 1 ) . a090418 . fromInteger) a000040_list
-- Reinhard Zumkeller, Aug 06 2012

PARI

is_A090423(n)={isprime(n)&&for(i=2, #binary(n)-2, bittest(n, i-1)&&isprime(n%2^i)&&is_A090421(n>>i)&&return(1))} \\ M. F. Hasler, Apr 21 2015

Python

# Primes = [2,...,607]
from sympy import sieve
primes = list(sieve.primerange(1, 608))
def tryPartioning(binString):   # First digit is not 0
    l = len(binString)
    for t in range(2, l-1):
        substr1 = binString[:t]
        if (int('0b'+substr1,2) in primes) or (t>=4 and tryPartioning(substr1)):
            substr2 = binString[t:]
            if substr2[0]!='0':
                if (int('0b'+substr2,2) in primes) or (l-t>=4 and tryPartioning(substr2)):
                    return 1
    return 0
for p in primes:
    if tryPartioning(bin(p)[2:]):
        print(p, end=',')

Python

from sympy import isprime, primerange
def ok(p):
  b = bin(p)[2:]
  for i in range(2, len(b)-1):
    if isprime(int(b[:i], 2)) and b[i] != '0':
      if isprime(int(b[i:], 2)) or ok(int(b[i:], 2)): return True
  return False
def aupto(lim): return [p for p in primerange(2, lim+1) if ok(p)]
print(aupto(607)) # Michael S. Branicky, May 16 2021

Extensions

Corrected by Alex Ratushnyak, Aug 03 2012