A053186 Square excess of n: difference between n and largest square <= n.

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

Offset

0,4

Comments

From David W. Wilson, Jan 05 2009: (Start)

More generally we may consider sequences defined by:

a(n) = n^j - (largest k-th power <= n^j),

a(n) = n^j - (largest k-th power < n^j),

a(n) = (largest k-th power >= n^j) - n^j,

a(n) = (largest k-th power > n^j) - n^j,

for small values of j and k.

The present entry is the first of these with j = 1 and n = 2.

It might be interesting to add further examples to the OEIS. (End)

a(A000290(n)) = 0; a(A005563(n)) = 2*n. - Reinhard Zumkeller, May 20 2009

0 ^ a(n) = A010052(n). - Reinhard Zumkeller, Feb 12 2012

From Frank M Jackson, Sep 21 2017: (Start)

The square excess of n has a reference in the Bakhshali Manuscript of Indian mathematics elements of which are dated between AD 200 and 900. A section within describes how to estimate the approximate value of irrational square roots. It states that for n an integer with an irrational square root, let b^2 be the nearest perfect square < n and a (=a(n)) be the square excess of n, then

sqrt(n) = sqrt(b^2+a) ~ b + a/(2b) - (a/(2b))^2/(2(b+a/(2b))). (End)

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

H. Bottomley, Illustration of A000196, A048760, A053186

J. J. O'Connor, E. F. Robertson.The Bakhshali manuscript, Historical Topics, St Andrews University.

Michael Somos, Sequences used for indexing triangular or square arrays

S. H. Weintraub, An interesting recursion, Amer. Math. Monthly, 111 (No. 6, 2004), 528-530.

Wikipedia, Bakhshali manuscript

Formula

a(n) = n - A048760(n) = n - floor(sqrt(n))^2.

a(n) = f(n,1) with f(n,m) = if n < m then n else f(n-m,m+2). - Reinhard Zumkeller, May 20 2009

Maple

A053186 := proc(n) n-(floor(sqrt(n)))^2 ; end proc;

Mathematica

f[n_] := n - (Floor@ Sqrt@ n)^2; Table[f@ n, {n, 0, 94}] (* Robert G. Wilson v, Jan 23 2009 *)

Prog

(PARI) A053186(n)= { if(n<0, 0, n-sqrtint(n)^2) }
(Haskell)
a053186 n = n - a048760 n
a053186_list = f 0 0 (map fst $ iterate (\(y, z) -> (y+z, z+2)) (0, 1))
   where f e x ys'@(y:ys) | x < y  = e : f (e + 1) (x + 1) ys'
                          | x == y = 0 : f 1 (x + 1) ys
-- Reinhard Zumkeller, Apr 27 2012

Crossrefs

Cf. A002262, A048760. A071797(n) = 1 + a(n-1).

Cf. A002262. - Reinhard Zumkeller, May 20 2009

Cf. A048760, A000196.

Sequence in context: A241382 A049260 A273294 * A066628 A255120 A218601

Adjacent sequences: A053183 A053184 A053185 * A053187 A053188 A053189

Keyword

easy,nonn

Author

Henry Bottomley, Mar 01 2000

Status

approved