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

1,1

Comments

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

Links

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

Example

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.

Prog

(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
(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

Crossrefs

Cf. A090422, A000040, A004676, A007088.

Sequence in context: A122259 A157173 A257318 * A232085 A086102 A058340

Adjacent sequences: A090420 A090421 A090422 * A090424 A090425 A090426

Keyword

nonn,base

Author

Reinhard Zumkeller, Nov 30 2003

Extensions

Corrected by Alex Ratushnyak, Aug 03 2012

Status

approved