cp's OEIS Frontend

A067603 Least k such that gcd(prime(k)+1, prime(k+1)+1) = 2n.

%I A067603 #40 Jan 08 2025 05:11:23
%S A067603 2,4,9,72,34,91,62,478,205,2016,522,909,1440,5375,2149,6610,7604,2976,
%T A067603 5229,7488,11251,7499,8805,20179,18526,70885,28193,40985,33847,17625,
%U A067603 27069,77199,66156,90764,26186,141235,70317,856719,110769,50523,217229
%N A067603 Least k such that gcd(prime(k)+1, prime(k+1)+1) = 2n.
%C A067603 Since all consecutive primes, 2 < p < q, are odd, therefore gcd(p+1, q+1) must be even.
%H A067603 Zak Seidov, Robert G. Wilson v, and Charles R Greathouse IV, <a href="/A067603/b067603.txt">Table of n, a(n) for n = 1..200</a> (1..100 terms from Seidov, 101..140 from Wilson, 141..200 from Greathouse)
%F A067603 Conjecture: a(n) = least k such that A001223(k) = 2n and A000040(k) == -1 (mod 2n). - _Zak Seidov_, Aug 16 2015
%e A067603 a(1) = 2, the first entry in A066940,
%e A067603 a(2) = 4, the first entry in A066941,
%e A067603 a(3) = 9, the first entry in A066942,
%e A067603 a(4) = 72, the first entry in A066943,
%e A067603 a(5) = 34, the first entry in A066944.
%e A067603 That is to say that the first k-th prime that has gcd(prime(k+1)+1, prime(k)+1) of 2, 4, 6, 8, 10, ..., are k = 2, 4, 9, 72, 34, ..., and the prime_k = 3, 7, 23, 359, 139, 467, 293, ... (A067604).
%e A067603 If the floor of GCD is used, then a(0) equals 1.
%t A067603 t = 0*Range@ 70; p = 3; q = 5; While[p < 15*10^6, d = GCD[p + 1, q + 1]/2; If[ t[[d]] == 0, t[[d]] = PrimePi@ p]; p = q; q = NextPrime@ q]; t
%o A067603 (PARI) a(n) = p=2; q=3; k=1; while(gcd(p+1, q+1) != 2*n, k++; p=q; q = nextprime(p+1);); k; \\ _Michel Marcus_, Aug 16 2015
%o A067603 (PARI) a(n)=my(p=2,k=2*n,t); forprime(q=3,, t++; if((q-p)%k==0 && (p+1)%k==0, return(t)); p=q) \\ _Charles R Greathouse IV_, Aug 17 2015
%o A067603 (PARI) a(n)=my(k=2*n); forstep(p=k-1,oo,k, if(isprime(p) && (nextprime(p+1)-p)%k==0, return(primepi(p)))) \\ _Charles R Greathouse IV_, Aug 17 2015
%o A067603 (MATLAB)
%o A067603 P = primes(10^8);
%o A067603 G = gcd(P(1:end-1)+1,P(2:end)+1);
%o A067603 A = zeros(1,66);
%o A067603 for n = 1:66
%o A067603     A(n) = find(G == 2*n, 1, 'first');
%o A067603 end
%o A067603 A   % _Robert Israel_, Aug 16 2015
%Y A067603 Main entry is A067604. Cf. A066940, A066941, A066942, A066943, A066944, A000040, A001223.
%K A067603 nonn
%O A067603 1,1
%A A067603 _Robert G. Wilson v_, Jan 31 2002
%E A067603 Edited by _Robert G. Wilson v_, Aug 17 2015 at the direction of _Zak Seidov_