A232091 Smallest square or promic (oblong) number greater than or equal to n.

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, 30, 30, 30, 30, 30, 36, 36, 36, 36, 36, 36, 42, 42, 42, 42, 42, 42, 49, 49, 49, 49, 49, 49, 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

Offset: 0

Author

L. Edson Jeffery, Nov 18 2013

Keywords

• nonn
• easy

Comments

Result attributed to the students Daring, et al., in the links section.

a(n) appears in floor(sqrt(a(n))) = A000194(n) successive terms.

Counting successive equal terms give sequence: 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ... (see A008619). - Michel Marcus, Jan 10 2014

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
• David Applegate, Proof of the equality A216607(n) = A232091(n) - n.
• E. Daring, I. Guadarrama, S. Sprague, and C. Winterer, WhaleConjecture.
• Casey Douglas, The Next Square or Pronic, June 2012. [Wayback Machine copy]

Crossrefs

Cf. A048761, A235382, A259225.

Cf. A000290 (squares), A002378 (promic or oblong numbers), A002620 (A000290 union A002378).

Programs

Magma

[(Ceiling(n /Ceiling(Sqrt(n)))*Ceiling(Sqrt(n))): n in [1..80]]; // Vincenzo Librandi, Jun 22 2015

Mathematica

Join[{0}, Table[Ceiling[n/Ceiling[Sqrt[n]]] Ceiling[Sqrt[n]], {n, 100}]] (* Alonso del Arte, Nov 18 2013 *)

PARI

a(n)=my(t=sqrtint(n-1)+1);t*((n-1)\t+1) \\ Charles R Greathouse IV, Nov 18 2013

Formula

a(n) = ceiling(n/ceiling(sqrt(n)))*ceiling(sqrt(n)).

a(n) = min(k : k >= n, k in A002620).

a(k^2) = k^2; a(k*(k+1)) = k*(k+1).

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]).

a(n) = min(A048761(n), A259225(n)). - Michel Marcus, Jun 22 2015

Sum_{n>=1} 1/a(n)^2 = 2 - Pi^2/6 + zeta(3). - Amiram Eldar, Aug 16 2022

Extensions

Extended by Charles R Greathouse IV, Nov 18 2013

a(0)=0 prepended by Michel Marcus, Jun 22 2015