A088149 Smallest prime which when rotated through all its binary places produces n primes, not counting any repeats.

2, 5, 11, 43, 167, 2143, 2423, 2687, 41903, 548543, 711679, 786431, 9010423, 10452461, 10065788911, 34762189783, 37974998491, 39115947389

Offset

1,1

Comments

It is probably not the case that this always produces the same bit cycle as A088148. - Franklin T. Adams-Watters, Mar 29 2014

Links

Table of n, a(n) for n=1..18.

Example

a(5) = 167 because 167 in base two is 10100111. This will produce eight possible new numbers; 01001111 = 79, 10011110 = 158, 00111101 = 61, 01111010 = 122, 11110100 = 244, 11101001 = 233, 11010011 = 211 and back to the beginning 10100111 = 167. Of those eight numbers (79, 158, 61, 122, 244, 233, 211 & 167) only five of them are primes. Notice that this is the same bit cycle as in A088148 but rotated differently.

Mathematica

f[n_] := Count[ PrimeQ[ Union[ Table[ FromDigits[ RotateLeft[ IntegerDigits[n, 2], i], 2], {i, 1, Floor[ Log[2, n] + 1]}]]], True]; NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; a = Table[0, {100}]; k = 2; Do[c = f[k]; If[c < 101 && a[[c]] == 0, a[[c]] = k]; k = NextPrim[k], {n, 1, 2750000}]; a

Prog

(Python 3.10+)
from itertools import count
from sympy import isprime
def A088149(n):
    if n == 1: return 2
    for p in count((1<<n)-1, 2):
        if p.bit_count() >= n and isprime(p):
            m = p.bit_length()
            l = 1<<m-1
            k, cset, q = l-1, {p}, p
            for _ in range(m-1):
                p = bool(p&l)+((p&k)<<1)
                if p not in cset and isprime(p):
                    cset.add(p)
            if len(cset) == n:
                return q # Chai Wah Wu, Jan 23 2023

Crossrefs

Cf. A088148.

Sequence in context: A172297 A128231 A088148 * A153989 A160859 A106887

Adjacent sequences: A088146 A088147 A088148 * A088150 A088151 A088152

Keyword

nonn,base,more

Author

Robert G. Wilson v, Sep 19 2003

Extensions

Edited by Franklin T. Adams-Watters, Mar 29 2014

a(15) from Chai Wah Wu, Jan 23 2023

a(16) from Chai Wah Wu, Jan 24 2023

a(17)-a(18) from Chai Wah Wu, Jan 25 2023

Status

approved