A243916 -id:A243916 - cp's OEIS frontend

A057821 a(n) is the least nonnegative integer k such that 2^n - k is a safe prime.

1, 5, 9, 5, 21, 29, 9, 5, 9, 17, 45, 161, 165, 269, 285, 17, 45, 233, 9, 17, 321, 317, 633, 677, 405, 437, 189, 1385, 69, 209, 9, 641, 849, 137, 45, 401, 381, 437, 1965, 2201, 741, 1493, 573, 857, 1485, 5297, 2709, 161, 465, 473, 1269, 4805, 789

Offset: 3

Warren D. Smith, Nov 23 2000

nonn

Previous name was: "Useful safe primes: a(n) = least nonnegative integer k such that 2^n - k is prime and (2^n-k-1)/2 is also prime". The resulting sequence of 2^n-k terms: 7, 11, 23, 59, 107, ..., are thus the largest safe primes smaller than 2^n (A243916), a subsequence of A005385. - Michel Marcus, Jan 08 2014

Mark Andreas, Table of n, a(n) for n = 3..5120 (first 3070 terms from Artsiom Palkounikau)

Cf. A005385, A013603, A243916.

a(n) = {my(k=0); until (isprime(2^n-k) && isprime((2^n-k-1)/2), k++); return (k);} \\ Michel Marcus, Jun 29 2013

from sympy import isprime
def a(n):
    k=0
    while True:
        k+=1
        if isprime(2**n - k) and isprime((2**n - k - 1)//2): return k
print([a(i) for i in range(3, 21)]) # Indranil Ghosh, Jun 12 2017, after PARI code by Michel Marcus

A350696 a(n) is the least nonnegative integer k such that 2^n + k is a safe prime.

Original entry on oeis.org

1, 3, 7, 15, 19, 39, 7, 51, 163, 15, 31, 231, 103, 75, 7, 195, 499, 99, 127, 627, 955, 555, 691, 87, 679, 99, 1411, 351, 799, 135, 91, 771, 79, 951, 667, 975, 1183, 1311, 667, 315, 955, 759, 2011, 9315, 4243, 1575, 907, 1527, 3943, 2091, 1927, 75, 1879

Offset: 2

Mark Andreas, Jan 12 2022

nonn

2^n+a(n) is the smallest (n+1)-bit safe prime.

			a(6)=19 because 2^6+19=83 is the smallest safe prime greater than 64 of the form p=2q+1 where p and q are both primes.

Mark Andreas, Table of n, a(n) for n = 2..5120

Cf. A111671, A057821, A243916, A181356, A335313.

safeQ[p_] := And @@ PrimeQ[{p, (p - 1)/2}]; a[n_] := Module[{k = 2^n + 1}, While[! safeQ[k], k++]; k -= 2^n]; Array[a, 50, 2] (* Amiram Eldar, Jan 12 2022 *)

a(n) = {my(k=0); until (isprime(2^n+k) && isprime((2^n+k-1)/2), k++); return (k); }

A345014 a(n) is the least nonnegative integer k such that 2^n - k is a Sophie Germain prime.

Original entry on oeis.org

0, 1, 3, 5, 3, 11, 15, 5, 3, 5, 9, 23, 81, 83, 135, 143, 9, 23, 117, 5, 9, 161, 159, 317, 339, 203, 219, 95, 693, 35, 105, 5, 321, 425, 69, 23, 201, 191, 219, 983, 1101, 371, 747, 287, 429, 743, 2649, 1355, 81, 233, 237, 635, 2403, 395, 1125, 1997, 69, 9005

Offset: 1

Artsiom Palkounikau, Sep 15 2021

nonn

Artsiom Palkounikau, Table of n, a(n) for n = 1..3072

Cf. A005384, A057821, A013603, A243916.

Table[k=0;While[!(PrimeQ[p=2^n-k]&&PrimeQ[2p+1]),k++];k,{n,58}] (* Giorgos Kalogeropoulos, Sep 15 2021 *)

a(n) = my(k=0,p); while (!(isprime(p=2^n-k) && isprime(2*p+1)), k++); k; \\ Michel Marcus, Sep 15 2021

from sympy import isprime
def a(n):
    k = 0
    while True:
        if isprime(2 ** n - k) and isprime(2 * (2 ** n - k) + 1):
            return k
        k += 1
print([a(i) for i in range(1, 21)])

a(n) = (A057821(n+1) + 1)/2.

cp's OEIS Frontend

A057821 a(n) is the least nonnegative integer k such that 2^n - k is a safe prime.

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A350696 a(n) is the least nonnegative integer k such that 2^n + k is a safe prime.

Views

Author

Keywords

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Programs

Mathematica

PARI

A345014 a(n) is the least nonnegative integer k such that 2^n - k is a Sophie Germain prime.

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula