A257230 Floor(sqrt(q)-(q-p)), where p and q are consecutive primes.

0, 0, 0, -1, 1, 0, 2, 0, -1, 3, 0, 2, 4, 2, 1, 1, 5, 2, 4, 6, 2, 5, 3, 1, 6, 8, 6, 8, 6, -3, 7, 5, 9, 2, 10, 6, 6, 8, 7, 7, 11, 3, 11, 10, 12, 2, 2, 11, 13, 11, 9, 13, 5, 10, 10, 10, 14, 10, 12, 14, 7, 3, 13, 15, 13, 4, 12, 8, 16, 14, 12, 11, 13, 13, 15, 13, 11, 16, 12, 10, 18

Offset

1,7

Comments

Conjecture: a(n) is always positive for n > 30, and is negative only for n = 4, 9 and 30, corresponding to prime pairs (7, 11), (23, 29) and (113, 127).

Related to prime gap conjectures by (e.g.) Legendre, Oppermann, Andrica and Brocard.

Links

Chris Boyd, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Prime Gaps

Wikipedia, Legendre's conjecture

Example

a(30) = -3 because sqrt(127)-(127-113) = -2.73057...
a(31) = 7 because sqrt(131)-(131-127) = 7.44552...

Mathematica

Table[Floor[Sqrt[NextPrime[Prime@ p]] - (NextPrime[Prime@ p] - Prime@ p)], {p, 81}] (* Michael De Vlieger, Apr 19 2015 *)

Prog

(PARI) a(n)=floor(sqrt(prime(n+1))-(prime(n+1)-prime(n)))
(Magma) [Floor(Sqrt(NthPrime(n+1))-(NthPrime(n+1)-NthPrime(n))): n in [1..100]]; // Vincenzo Librandi, Apr 19 2015

Crossrefs

Cf. A257231.

Sequence in context: A353778 A173662 A263406 * A357136 A172026 A296046

Adjacent sequences: A257227 A257228 A257229 * A257231 A257232 A257233

Keyword

sign,easy

Author

Chris Boyd, Apr 19 2015

Status

approved