A067603 Least k such that gcd(prime(k)+1, prime(k+1)+1) = 2n.
2, 4, 9, 72, 34, 91, 62, 478, 205, 2016, 522, 909, 1440, 5375, 2149, 6610, 7604, 2976, 5229, 7488, 11251, 7499, 8805, 20179, 18526, 70885, 28193, 40985, 33847, 17625, 27069, 77199, 66156, 90764, 26186, 141235, 70317, 856719, 110769, 50523, 217229
Offset: 1
Keywords
Examples
a(1) = 2, the first entry in A066940, a(2) = 4, the first entry in A066941, a(3) = 9, the first entry in A066942, a(4) = 72, the first entry in A066943, a(5) = 34, the first entry in A066944. 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). If the floor of GCD is used, then a(0) equals 1.
Links
- Zak Seidov, Robert G. Wilson v, and Charles R Greathouse IV, Table of n, a(n) for n = 1..200 (1..100 terms from Seidov, 101..140 from Wilson, 141..200 from Greathouse)
Programs
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MATLAB
P = primes(10^8); G = gcd(P(1:end-1)+1,P(2:end)+1); A = zeros(1,66); for n = 1:66 A(n) = find(G == 2*n, 1, 'first'); end A % Robert Israel, Aug 16 2015 -
Mathematica
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
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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
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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
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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
Formula
Conjecture: a(n) = least k such that A001223(k) = 2n and A000040(k) == -1 (mod 2n). - Zak Seidov, Aug 16 2015
Extensions
Edited by Robert G. Wilson v, Aug 17 2015 at the direction of Zak Seidov
Comments