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A078444
Floor of geometric mean of two consecutive primes.
3
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
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
Sequence in context: A301892 A271876 A358533 * A332071 A225087 A194221
KEYWORD
nonn,easy
AUTHOR
Lior Manor, Dec 31 2002
STATUS
approved