A035096 -id:A035096 - cp's OEIS frontend

A357746 Primes p such that the least k for which kp + 1 is prime is also the least k for which kp - 1 is prime.

47, 103, 107, 283, 313, 347, 397, 773, 787, 907, 1051, 1117, 1319, 1433, 1823, 2027, 2153, 2203, 2287, 2333, 2347, 2381, 2909, 3221, 3257, 3673, 3923, 3929, 4129, 4153, 4217, 4547, 4597, 4657, 4721, 4969, 5023, 5387, 5407, 5693, 5717, 5827, 5881, 6373, 6781, 6863, 6997

Offset: 1

Karl-Heinz Hofmann, Jan 01 2023

nonn

If A035096(n) = A216568(n) the n-th prime is a term. Here k*p must be the composite number sandwiched between a pair of twin primes, so by Wilson's theorem, k must be a multiple of 6.

			a(1) = 47: 47*6 + 1 = 283 (a prime), 47*6 - 1 = 281 (also a prime), and no k < 6 gives a prime as the result for both formulas.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000
Wikipedia, Wilson's theorem.

Cf. A000040, A014574, A001359, A006512, A035096, A216568.

q[p_] := Module[{k = 1, r}, While[! Or @@ (r = PrimeQ[k*p + {-1, 1}]), k++]; And @@ r]; Select[Prime[Range[900]], q] (* Amiram Eldar, Jan 01 2023 *)

isk(p, x) = my(k=1); while (!isprime(k*p+x), k++); k;
isok(p) = if (isprime(p), isk(p, +1) == isk(p, -1)); \\ Michel Marcus, Jan 01 2023

from sympy import sieve, isprime
def leastk(p, plusminus):
    k=1
    while not isprime(k * p + plusminus): k += 1
    return k
print([p for p in sieve[1:1000] if leastk(p, 1) == leastk(p, -1)])

A367805 a(1) = 0; for n > 1, a(n) is the least positive integer k for which k*prime(n) + 2 is prime.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 1, 3, 3, 1, 5, 3, 1, 3, 7, 7, 1, 5, 5, 1, 5, 3, 3, 3, 3, 1, 3, 1, 5, 9, 3, 7, 1, 3, 1, 5, 5, 3, 3, 3, 1, 5, 1, 5, 1, 3, 9, 5, 1, 9, 3, 1, 15, 7, 3, 15, 1, 9, 11, 1, 9, 3, 21, 1, 3, 3, 5, 3, 1, 3, 3, 15, 3, 5, 9, 3, 13, 3, 19, 3, 1, 15, 1, 3, 3, 9, 13, 3, 1, 15

Offset: 1

Frank Hollstein, Dec 01 2023

nonn

			For n = 4: a(4) = 3, because prime(4) = 7, 3*7 + 2 = 23 which is prime.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000040, A029707, A035096, A117673, A279756, A001359.

a:= proc(n) local p, q, r; p:= ithprime(n); q:= p;
      while irem(q-2, p, 'r')<>0 do q:= nextprime(q) od; r
    end:
seq(a(n), n=1..99);  # Alois P. Heinz, Dec 04 2023

nmax=90; a[1]=0; For[n=2, n<=nmax, n++, For[k=1, k>0, k++, If[PrimeQ[k*Prime[n]+2], a[n]=k; k=-1]]]; Array[a,nmax] (* Stefano Spezia, Dec 04 2023 *)

a(n) = if (n==1, 0, my(k=1, p=prime(n)); while (!isprime(k*p+2), k++); k); \\ Michel Marcus, Dec 02 2023

from itertools import count, dropwhile
from sympy import prime, isprime
def A367805(n):
    if n==1:
        return 0
    else:
        p = prime(n)
        return next(dropwhile(lambda x:not isprime(x*p+2),count(1))) # Chai Wah Wu, Jan 04 2024

a(n) = (A279756(n) - 2)/A000040(n).

a(n) = 1 <=> n in A029707.

More terms from Michel Marcus, Dec 02 2023

A266239 a(n) is the smallest k such that the following are four primes: prime(n)k-1, prime(n)k+1, prime(n)k^2-1, prime(n)k^2+1. Or -1 if no such k exists.

Original entry on oeis.org

3, 2, 6, 66, 300, 24, 3744, 27000, 6, 1080, 960, 90, 30, 366, 9204, 8154, 1170, 840, 330, 6090, 84, 27720, 324, 90, 186, 4740, 2856, 6, 24270, 936, 5220, 1560, 6, 1500, 510, 120, 930, 3270, 30, 8520, 29700, 1200, 23310, 1056, 3540, 90, 3420, 2370, 9654, 83040, 2610, 9270, 2610

Offset: 1

Alex Ratushnyak, Dec 25 2015

nonn

Conjecture: a(n)>1.

			a(1)=3 because the following are four primes: 2*3-1, 2*3+1, 2*9-1, 2*9+1.

Cf. A000040, A035096.

a(n) = {k = 1; p = prime(n); while (! (isprime(p*k-1) && isprime(p*k+1) && isprime(p*k^2-1) && isprime(p*k^2+1)), k++);k;} \\ Michel Marcus, Dec 27 2015

from sympy import isprime
TOP=10**9
for p in range(2, 333):
    if isprime(p):
        failed = True
        for x in range(1, TOP):
            if isprime(p*x+1) and isprime(p*x-1) and isprime(p*x*x-1) and isprime(p*x*x+1):
                print(x, end=', ')
                failed = False
                break
        if failed: print(-1, end=', ')

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A357746 Primes p such that the least k for which kp + 1 is prime is also the least k for which kp - 1 is prime.

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A367805 a(1) = 0; for n > 1, a(n) is the least positive integer k for which k*prime(n) + 2 is prime.

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A266239 a(n) is the smallest k such that the following are four primes: prime(n)k-1, prime(n)k+1, prime(n)k^2-1, prime(n)k^2+1. Or -1 if no such k exists.

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cp's OEIS Frontend

A357746 Primes p such that the least k for which k*p + 1 is prime is also the least k for which k*p - 1 is prime.

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A367805 a(1) = 0; for n > 1, a(n) is the least positive integer k for which k*prime(n) + 2 is prime.

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Maple

Mathematica

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Extensions

A266239 a(n) is the smallest k such that the following are four primes: prime(n)*k-1, prime(n)*k+1, prime(n)*k^2-1, prime(n)*k^2+1. Or -1 if no such k exists.

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PARI

Python

A357746 Primes p such that the least k for which kp + 1 is prime is also the least k for which kp - 1 is prime.

A266239 a(n) is the smallest k such that the following are four primes: prime(n)k-1, prime(n)k+1, prime(n)k^2-1, prime(n)k^2+1. Or -1 if no such k exists.