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

%I A067605 #20 Mar 05 2025 01:39:21
%S A067605 2,6,11,24,42,121,30,319,99,1592,344,574,3786,4196,650,4619,217,1532,
%T A067605 11244,5349,8081,3861,12751,18281,9221,5995,22467,16222,43969,35975,
%U A067605 192603,108146,52313,218234,15927,132997,42673,78858,103865,84483,111172,175288,110734
%N A067605 Least k such that gcd(prime(k+1)-1, prime(k)-1) = 2n.
%C A067605 Since all consecutive primes, p < q and p greater than 2, are odd, therefore gcd(p-1, q-1) must be even.
%H A067605 Amiram Eldar, <a href="/A067605/b067605.txt">Table of n, a(n) for n = 1..183</a>
%F A067605 a(n) = PrimePi(A058264(n)).
%e A067605 For n = 4: a(4) = 24 = gcd(89-1, 97-1) = gcd(p(24)-1, p(25)-1) = 8 = 2*4.
%p A067605 N:= 50: # for a(1)..a(N)
%p A067605 V:= Vector(N): count:= 0:
%p A067605 p:= 3:
%p A067605 for k from 2 while count < N do
%p A067605   q:= p;
%p A067605   p:= nextprime(p);
%p A067605   v:= igcd(p-1,q-1)/2;
%p A067605   if v <= N and V[v] = 0 then
%p A067605     count:= count+1; V[v]:= k;
%p A067605   fi
%p A067605 od:
%p A067605 convert(V,list); # _Robert Israel_, Mar 05 2025
%t A067605 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
%o A067605 (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
%Y A067605 Cf. A000720, A063444, A084307, A058263, A058264.
%K A067605 nonn
%O A067605 1,1
%A A067605 _Robert G. Wilson v_, Jan 31 2002