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A333245 Primes p such that the order of 2 mod p is less than the square root of p. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified April 16 18:53 EDT 2021. Contains 343050 sequences. (Running on oeis4.)