A067605 Least k such that gcd(prime(k+1)-1, prime(k)-1) = 2n.

2, 6, 11, 24, 42, 121, 30, 319, 99, 1592, 344, 574, 3786, 4196, 650, 4619, 217, 1532, 11244, 5349, 8081, 3861, 12751, 18281, 9221, 5995, 22467, 16222, 43969, 35975, 192603, 108146, 52313, 218234, 15927, 132997, 42673, 78858, 103865, 84483, 111172, 175288, 110734

Offset: 1

Author

Robert G. Wilson v, Jan 31 2002

Keywords

• nonn

Comments

Since all consecutive primes, p < q and p greater than 2, are odd, therefore gcd(p-1, q-1) must be even.

Examples

			For n = 4: a(4) = 24 = gcd(89-1, 97-1) = gcd(p(24)-1, p(25)-1) = 8 = 2*4.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..183

Crossrefs

Cf. A000720, A063444, A084307, A058263, A058264.

Programs

Maple

N:= 50: # for a(1)..a(N)
V:= Vector(N): count:= 0:
p:= 3:
for k from 2 while count < N do
  q:= p;
  p:= nextprime(p);
  v:= igcd(p-1,q-1)/2;
  if v <= N and V[v] = 0 then
    count:= count+1; V[v]:= k;
  fi
od:
convert(V,list); # Robert Israel, Mar 05 2025

Mathematica

a = Table[0, {100}]; p = 3; q = 5; Do[q = Prime[n + 1]; d = GCD[p - 1, q - 1]/2; If[d < 101 && a[[d]] == 0, a[[d]] = n]; b = c, {n, 2, 10^7}]; a

PARI

list(len) = {my(v = vector(len), c = 0, p = 3, k = 2, i); forprime(q = 5, , i = gcd(p-1, q-1)/2; if(i <= len && v[i] == 0, v[i] = k; c++; if(c == len, break)); p = q; k++); v;} \\ Amiram Eldar, Mar 05 2025

Formula

a(n) = PrimePi(A058264(n)).