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)
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
KEYWORD
easy,nonn
AUTHOR
Henry Bottomley, Mar 01 2000
STATUS
approved