

A061265


Number of squares between nth prime and (n+1)st prime.


9



0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1
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OFFSET

1,1


COMMENTS

If nth prime is a member of A053001 then a(n) is at least 1. If not, then a(n) = 0.
Legendre's conjecture (still open) that there is always a prime between n^2 and (n+1)^2 is equivalent to conjecturing that a(n) <= 1 for all n.  Vladeta Jovovic, May 01 2003
a(A038107(n)) = 1 for n > 1; a(A221056(n)) = 0.  Reinhard Zumkeller, Apr 15 2013


LINKS

Harry J. Smith, Table of n, a(n) for n = 1..2000
Eric Weisstein's World of Mathematics, Legendre's Conjecture
Wikipedia, Legendre's conjecture


FORMULA

a(n) = floor(sqrt(prime(n+1)))  floor(sqrt(prime(n))).  Vladeta Jovovic, May 01 2003


EXAMPLE

a(3) = 0 as there is no square between 5, the third prime and 7, the fourth prime. a(4) = 1, as there is a square (9) between the 4th prime 7 and the 5th prime 11.


MATHEMATICA

ns[{a_, b_}]:=Count[Range[a+1, b1], _?(IntegerQ[Sqrt[#]]&)]; ns/@ Partition[ Prime[Range[110]], 2, 1] (* Harvey P. Dale, Mar 14 2015 *)


PROG

(PARI) { n=0; q=2; forprime (p=3, prime(2001), write("b061265.txt", n++, " ", floor(sqrt(p))floor(sqrt(q))); q=p ) } \\ Harry J. Smith, Jul 20 2009
(Haskell)
a061265 n = a061265_list !! (n1)
a061265_list = map sum $
zipWith (\u v > map a010052 [u..v]) a000040_list $ tail a000040_list
 Reinhard Zumkeller, Apr 15 2013


CROSSREFS

Cf. A053001.
Cf. A038107.
Cf. A014085.
Sequence in context: A268384 A288524 A112416 * A288466 A285073 A276394
Adjacent sequences: A061262 A061263 A061264 * A061266 A061267 A061268


KEYWORD

nonn,base


AUTHOR

Amarnath Murthy, Apr 24 2001


EXTENSIONS

Extended by Patrick De Geest, Jun 05 2001
Offset changed from 0 to 1 by Harry J. Smith, Jul 20 2009


STATUS

approved



