A086884 Smallest string of 1's and 0's which is prime in every base from 2 to n.

10, 10, 101111, 10010111, 111110100001, 111110100001, 11000011101101111, 10011110011011110110110011, 110100000010101111110001010011001110001, 1000000010000011110100010001000101001010110111001, 10100001011000101000110101011011011101111110100101011

Offset

2,1

Links

Table of n, a(n) for n=2..12.

Phil Carmody, Bitstrings representing primes in many bases.

Mersenneforum.org, Multibase primes

Example

a(5) = 10010111 because 10010111 (base 2) = 151, 10010111 (base 3) = 2281, 10010111 (base 4) = 16661 and 10010111 (base 5) = 78781 are all prime and 10010111 is the smallest such string.

Mathematica

a[n_] := (While[b = FromDigits[ IntegerDigits[k, 2]]; Union[ PrimeQ[ Table[ FromDigits[ IntegerDigits[b], i], {i, 2, n}]]] != {True}, k++ ]; b); k = 1; Do[ Print[ a[n]], {n, 2, 10}]

Prog

(Python)
from sympy import isprime
def conv(s, b): return sum(b**k for k, bk in enumerate(s[::-1]) if bk=='1')
def ok(s, n): return all(isprime(conv(s, b)) for b in range(2, n+1))
def a(n):
    if n < 4: return 10
    k = 3
    while not ok(bin(k)[2:], n): k += 2
    return int(bin(k)[2:])
print([a(n) for n in range(2, 9)]) # Michael S. Branicky, Oct 10 2021

Crossrefs

See A086204 for decimal equivalents.

Sequence in context: A052983 A072804 A079793 * A284201 A324660 A336035

Adjacent sequences: A086881 A086882 A086883 * A086885 A086886 A086887

Keyword

nonn,base

Author

Richard FitzHugh (fitzhughrichard(AT)hotmail.com), Aug 22 2003

Extensions

Edited by Robert G. Wilson v, Aug 24 2003

a(10) from Phil Carmody's Web site added by Dario Alpern, May 14 2006

a(11) from Phil Carmody's website added by James G. Merickel, Feb 15 2010

a(12) from Jim Viebke on mersenneforum.org added by Andreas Höglund, Oct 17 2022

Status

approved