A090422 Primes that cannot be written in binary representation as concatenation of other primes.

2, 3, 5, 7, 13, 17, 19, 37, 41, 53, 67, 73, 89, 97, 101, 103, 107, 131, 137, 139, 149, 163, 193, 197, 199, 211, 227, 257, 263, 269, 277, 281, 293, 307, 311, 313, 331, 389, 397, 401, 409, 419, 421, 443, 449, 461, 521, 523, 547, 557, 569, 571, 577, 587, 593

Offset

1,1

Comments

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

This sequence is indeed infinite, as we need infinitely many terms to cover the primes with arbitrarily large runs of 0's in their base-2 representation. - Jeffrey Shallit, Mar 07 2021

Links

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

Prog

(Haskell)
a090422 n = a090422_list !! (n-1)
a090422_list = filter ((== 1) . a090418 . fromInteger) a000040_list
-- Reinhard Zumkeller, Aug 07 2012
(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 not ok(int(b[i:], 2)): return False
  return True
def aupto(lim): return [p for p in primerange(2, lim+1) if ok(p)]
print(aupto(593)) # Michael S. Branicky, Mar 07 2021

Crossrefs

Cf. A090423, A000040, A004676, A007088.

A342244 handles the case where the primes are allowed to have leading zeros.

Sequence in context: A109461 A138539 A337119 * A005109 A247980 A234851

Adjacent sequences: A090419 A090420 A090421 * A090423 A090424 A090425

Keyword

nonn,base

Author

Reinhard Zumkeller, Nov 30 2003

Extensions

Based on corrections in A090418, data recomputed by Reinhard Zumkeller, Aug 07 2012

Status

approved