This site is supported by donations to The OEIS Foundation.

 Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS". Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A053186 Square excess of n: difference between n and largest square <= n. 37
 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 From Frank M Jackson, Sep 21 2019: (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. 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.