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Let k be the difference between the smallest square >= n and n. Sequence gives difference between the smallest square >= k and k.
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%I #28 Sep 08 2022 08:46:10

%S 0,2,0,0,0,1,2,0,0,3,4,0,1,2,0,0,1,2,3,4,0,1,2,0,0,6,0,1,2,3,4,0,1,2,

%T 0,0,4,5,6,0,1,2,3,4,0,1,2,0,0,2,3,4,5,6,0,1,2,3,4,0,1,2,0,0,0,1,2,3,

%U 4,5,6,0,1,2,3,4,0,1,2,0,0,7,8,0,1,2,3,4,5,6,0,1,2,3,4,0,1,2,0,0

%N Let k be the difference between the smallest square >= n and n. Sequence gives difference between the smallest square >= k and k.

%C Equals A068527 applied to itself.

%H G. C. Greubel, <a href="/A249142/b249142.txt">Table of n, a(n) for n = 1..5000</a>

%F a(n) = A068527(A068527(n)).

%F a(n) = n - ceiling(sqrt(n))^2 + ceiling(sqrt(-n+ceiling(sqrt(n))^2))^2.

%F a(n) < (64n)^(1/4). - _Charles R Greathouse IV_, Oct 22 2014

%e For n = 13 the next biggest square is 16, thus k = 16 - 13 = 3 and for 3 the next biggest square is 4, thus a(14) = 3 - 2 = 1.

%p A068527:= n -> ceil(sqrt(n))^2 - n:

%p map(A068527@@2, [$1..100]); # _Robert Israel_, Nov 02 2017

%t Table[n - Ceiling[Sqrt[n]]^2 + Ceiling[Sqrt[-n + Ceiling[Sqrt[n]]^2]]^2, {n, 1, 100}]

%o (PARI) A068527(n)=if(issquare(n), 0, (sqrtint(n)+1)^2-n)

%o a(n)=A068527(A068527(n)) \\ _Charles R Greathouse IV_, Oct 22 2014

%o (Magma) [n - Ceiling(Sqrt(n))^2 + Ceiling(Sqrt(-n+Ceiling(Sqrt(n))^2))^2: n in [1..100]]; // _Vincenzo Librandi_, Oct 23 20124

%Y Cf. A068527.

%K nonn,easy

%O 1,2

%A _Valtteri Raiko_, Oct 22 2014

%E Edited, old crossrefs entry moved to Comments, and first two formula lines interchanged by _Wolfdieter Lang_, Nov 10 2014