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

Author

Lior Manor, Dec 31 2002

Keywords

• nonn
• easy

Comments

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

Examples

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

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Andrica's Conjecture

Crossrefs

Cf. A001223, A006094.

Programs

Magma

[Floor(Sqrt(NthPrime(n)*NthPrime(n+1))): n in [1..60]]; // Vincenzo Librandi, Dec 12 2015

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 *)

PARI

a(n) = sqrtint(prime(n)*prime(n+1)); \\ Michel Marcus, Dec 12 2015

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