This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
t = {{1, 4}}; Do[k = 1; While[! (PrimeQ[k*n - 1] && PrimeQ[k*n + 1]), k++]; If[k > t[[-1, 2]], AppendTo[t, {n, k}]], {n, 2, 100000}]; Transpose[t][[2]]
a(4) = 2 because there are two twin prime pairs of the form (4k - 1,4k + 1) with 1 <= k <= 4, namely (3, 5) for k = 1 and (11, 13) for k = 3.
a[n_] := Sum[Boole[PrimeQ[k*n - 1] && PrimeQ[k*n + 1]], {k, 1, n}]; Array[a, 100] (* Amiram Eldar, Dec 25 2019 *)
a(n)=sum(k=1,n,isprime(k*n-1)&&isprime(k*n+1)) \\ Charles R Greathouse IV, Jun 27 2013
is(n)=for(k=1,n-1,if(isprime(n*k-1)&&isprime(n*k+1), return(0))); 1 \\ Charles R Greathouse IV, May 08 2015
is(n)=my(s=6/gcd(n,6)); forstep(k=s,n-1,s, if(isprime(n*k-1)&&isprime(n*k+1), return(0))); 1 \\ Charles R Greathouse IV, May 08 2015
Reap[Do[If[PrimeQ[a=3*2^n+5]&&PrimeQ[a+2],Sow[n]],{n,150}]][[2,1]]
for(k = 1,10^4, if(ispseudoprime(a = 3*2^k + 5)&&ispseudoprime (a + 2), print1(k",")))
a(1) = 4 because for k = 1, 1*(1^3) - 1 = 0 and 1*(1^3) + 1 = 2 are not twin primes, for k = 2, 1 and 3 are not twin primes, for k = 3, 2 and 4 are not twin primes, so the smallest k that works is k = 4: 4*(1^3) - 1 = 3 and 4*(1^3) + 1 = 5 are twin primes.
import sympy
from sympy import isprime
def f(n):
for k in range(1,10**4):
if isprime(k*(n**3)-1) and isprime(k*(n**3)+1):
return k
n = 1
while n < 10**3:
print(f(n))
n += 1
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