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

Author

Henry Bottomley, Mar 01 2000

Keywords

• easy
• nonn

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

Crossrefs

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

Cf. A002262. - Reinhard Zumkeller, May 20 2009

Cf. A048760, A000196.

Programs

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

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 *)

PARI

A053186(n)= { if(n<0,0,n-sqrtint(n)^2) }

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