A333245 Primes p such that the order of 2 mod p is less than the square root of p.

31, 127, 257, 683, 1103, 1801, 2089, 2113, 2351, 2731, 3191, 4051, 4513, 5419, 6361, 8191, 9719, 11119, 11447, 13367, 14449, 14951, 20231, 20857, 23279, 23311, 26317, 29191, 30269, 32377, 37171, 38737, 39551, 43441, 43691, 49477, 54001, 55633, 55871, 59393

Offset

1,1

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Example

The order of 2 mod 31 is 5, and sqrt(31) = 5.56776436283..., which is more than 5, so 31 is in the sequence.
The order of 2 mod 37 is 36, and sqrt(37) = 6.08276253..., which is significantly less than 36, so 37 is not in the sequence.

Maple

q:= p-> is(numtheory[order](2, p)^2<p):
select(q, [ithprime(i)$i=1..10000])[];  # Alois P. Heinz, Mar 16 2020

Mathematica

Select[Prime[Range[6000]], MultiplicativeOrder[2, #] < Sqrt[#] &] (* Amiram Eldar, Mar 16 2020 *)

Prog

(PARI) list(lim)=my(v=List(), t, p, o); forfactored(P=30, lim\1, if(vecsum(P[2][, 2])==1, t=znorder(Mod(2, p=P[1]), o); if(t^2<p, listput(v, p))); o=P); Vec(v)
(Julia)
using Nemo
function isA333245(n)
    ! isprime(n) && return false
    s, m, N = 0, 1, n
    r = isqrt(n)
    while true
        k = N + m
        v = valuation(k, 2)
        s += v
        s > r && return false
        m = k >> v
        m == 1 && break
    end
    return true
end
print([n for n in 3:2:60000 if isA333245(n)]) # Peter Luschny, Mar 16 2020

Crossrefs

Cf. A014664, A002326.

Sequence in context: A078656 A095322 A127578 * A158563 A079141 A049203

Adjacent sequences: A333242 A333243 A333244 * A333246 A333247 A333248

Keyword

nonn

Author

Charles R Greathouse IV, Mar 12 2020

Status

approved