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A053186 Square excess of n: difference between n and largest square <= n. 35
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 (list; graph; refs; listen; history; text; internal format)
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

LINKS

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

H. Bottomley, Illustration of A000196, A048760, A053186

M. Somos, Sequences used for indexing triangular or square arrays

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

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).

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

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Last modified August 21 13:36 EDT 2017. Contains 290890 sequences.