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A232091 Smallest square or promic (oblong) number greater than or equal to n. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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, C. Winterer, WhaleConjecture.

Casey Douglas, The Next Square or Pronic, June 2012.

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

MATHEMATICA

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

PROG

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

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

CROSSREFS

Cf. A235382.

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

Sequence in context: A211514 A111457 A161765 * A164798 A087554 A281072

Adjacent sequences:  A232088 A232089 A232090 * A232092 A232093 A232094

KEYWORD

nonn,easy

AUTHOR

L. Edson Jeffery, Nov 18 2013

EXTENSIONS

Extended by Charles R Greathouse IV, Nov 18 2013

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

STATUS

approved

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Last modified November 16 17:04 EST 2019. Contains 329201 sequences. (Running on oeis4.)