A078444 Floor of geometric mean of two consecutive primes.

2, 3, 5, 8, 11, 14, 17, 20, 25, 29, 33, 38, 41, 44, 49, 55, 59, 63, 68, 71, 75, 80, 85, 92, 98, 101, 104, 107, 110, 119, 128, 133, 137, 143, 149, 153, 159, 164, 169, 175, 179, 185, 191, 194, 197, 204, 216, 224, 227, 230, 235, 239, 245, 253, 259, 265, 269, 273, 278

Offset

1,1

Comments

For n > 1, a(n) = prime(n) iff prime(n) and prime(n+1) are twin primes.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Andrica's Conjecture

Formula

a(n) = floor(sqrt(prime(n)*prime(n+1))).

From Miko Labalan, Dec 12 2015: (Start)

a(n) = A006254(A028310(n - 1)) + A067076(n);

a(n) = A067076(A028310(n - 1)) + A006254(n);

a(n) = A005097(A028310(n - 1)) + A005097(n).

(End)

For n >= 2 these formulas are equivalent to sqrt(prime(n)*prime(n+1)) > (prime(n)+prime(n+1))/2 - 1, and thus to A001223(n) <= 2 + 2*sqrt(2*prime(n)). This would be implied by Andrica's conjecture, but is as yet unproven. - Robert Israel, Dec 13 2015

Example

a(7) = floor(sqrt(prime(7)*prime(8))) = 17.

Maple

seq(floor(sqrt(ithprime(i)*ithprime(i+1))), i=1..100); # Robert Israel, Dec 12 2015

Mathematica

Table[Floor[Sqrt[Prime[n] Prime[n + 1]]], {n, 60}] (* Vincenzo Librandi, Dec 12 2015 *)
Table[Ceiling[(Prime[n] + Prime[n + 1])/2 - 1], {n, 100}] (* Miko Labalan, Dec 14 2015 *)

Prog

(Magma) [Floor(Sqrt(NthPrime(n)*NthPrime(n+1))): n in [1..60]]; // Vincenzo Librandi, Dec 12 2015
(PARI) a(n) = sqrtint(prime(n)*prime(n+1)); \\ Michel Marcus, Dec 12 2015

Crossrefs

Cf. A001223, A006094.

Sequence in context: A301892 A271876 A358533 * A332071 A225087 A194221

Adjacent sequences: A078441 A078442 A078443 * A078445 A078446 A078447

Keyword

nonn,easy

Author

Lior Manor, Dec 31 2002

Status

approved