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%I #48 Jan 02 2023 12:30:49
%S 0,1,2,4,4,6,6,9,9,9,12,12,12,16,16,16,16,20,20,20,20,25,25,25,25,25,
%T 30,30,30,30,30,36,36,36,36,36,36,42,42,42,42,42,42,49,49,49,49,49,49,
%U 49,56,56,56,56,56,56,56,64,64,64,64,64,64,64,64,72,72,72,72,72,72,72,72,81
%N Smallest square or promic (oblong) number greater than or equal to n.
%C Result attributed to the students Daring, et al., in the links section.
%C a(n) appears in floor(sqrt(a(n)) = A000194(n) successive terms.
%C Counting successive equal terms give sequence: 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ... (see A008619). - _Michel Marcus_, Jan 10 2014
%H Charles R Greathouse IV, <a href="/A232091/b232091.txt">Table of n, a(n) for n = 0..10000</a>
%H David Applegate, <a href="http://list.seqfan.eu/oldermail/seqfan/2014-January/012229.html">Proof of the equality A216607(n) = A232091(n) - n</a>.
%H E. Daring, I. Guadarrama, S. Sprague, and C. Winterer, <a href="http://whaleconjecture.wordpress.com/">WhaleConjecture</a>.
%H Casey Douglas, <a href="https://web.archive.org/web/20150912213402/https://mathematicalypse.wordpress.com/2012/06/24/the-next-square-or-pronic/">The Next Square or Pronic</a>, June 2012. [Wayback Machine copy]
%F a(n) = ceiling(n/ceiling(sqrt(n)))*ceiling(sqrt(n)).
%F a(n) = min(k : k >= n, k in A002620).
%F a(k^2) = k^2; a(k*(k+1)) = k*(k+1).
%F It appears that a(n) = A216607(n) + n. (Verified for all n<10^9 by _Lars Blomberg_, Jan 09 2014.) This conjecture now follows from a proof given by _David Applegate_, Jan 10 2014 (see [Applegate]).
%F a(n) = min(A048761(n), A259225(n)). - _Michel Marcus_, Jun 22 2015
%F Sum_{n>=1} 1/a(n)^2 = 2 - Pi^2/6 + zeta(3). - _Amiram Eldar_, Aug 16 2022
%t Join[{0}, Table[Ceiling[n/Ceiling[Sqrt[n]]] Ceiling[Sqrt[n]], {n, 100}]] (* _Alonso del Arte_, Nov 18 2013 *)
%o (PARI) a(n)=my(t=sqrtint(n-1)+1);t*((n-1)\t+1) \\ _Charles R Greathouse IV_, Nov 18 2013
%o (Magma) [(Ceiling(n /Ceiling(Sqrt(n)))*Ceiling(Sqrt(n))): n in [1..80]]; // _Vincenzo Librandi_, Jun 22 2015
%Y Cf. A048761, A235382, A259225.
%Y Cf. A000290 (squares), A002378 (promic or oblong numbers), A002620 (A000290 union A002378).
%K nonn,easy
%O 0,3
%A _L. Edson Jeffery_, Nov 18 2013
%E Extended by _Charles R Greathouse IV_, Nov 18 2013
%E a(0)=0 prepended by _Michel Marcus_, Jun 22 2015
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