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A080673 Largest number with exactly n representations as sum of five positive squares (or 0 if no number with exactly n representations exists). 4
33, 60, 105, 90, 132, 177, 145, 201, 225, 180, 297, 228, 213, 265, 345, 258, 305, 300, 393, 369, 465, 417, 385, 337, 420, 425, 513, 537, 409, 473, 449, 585, 561, 545, 481, 505, 633, 540, 528, 705, 593, 462, 665, 681, 617, 825, 564, 753, 548, 689, 777, 713 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

a(0) = 33 has been asked as a riddle by Gerhard Woeginger in de.rec.denksport.

There is no number <= 10^6 that is the sum of five positive squares in exactly 188 ways. - Donovan Johnson, Aug 15 2013

Donovan Johnson's exhaustive search in fact shows that a(188) is undefined / should be assigned the default value 0. Hagen von Eitzen, Jun 05 2014

In what sense the search is exhaustive? Is any of the  0's of the b-file proved? - M. F. Hasler, Oct 27 2017

LINKS

Hagen von Eitzen, Table of n, a(n) for n = 0..39054 [This replaces an earlier b-file computed by Donovan Johnson]

H. v. Eitzen, in reply to user James47, What is the largest integer with only one representation as a sum of five nonzero squares?, on stackexchange.com, May 2014

FORMULA

a(n) = max { k | A025429(k) = n }. - M. F. Hasler, May 30 2014

EXAMPLE

A known result says a(0) = 33, since there is no representation of 33 as sum of 5 positive squares.

a(1) = 60 is the largest number allowing exactly one such representation.

MATHEMATICA

max = 1000; m = Ceiling[Sqrt[max]]; xx = Array[x, 6, 0]; x[0] = 1; iter = Sequence @@ Thread[{Rest[xx], Most[xx], m}]; representations = Table[ Rest[xx] . Rest[xx], Evaluate[iter]] // Flatten // Sort // Split // Select[#, First[#] <= max &] &; counts = {First[#], Length[#]} & /@ representations; a[0] = Complement[Range[max], counts[[All, 1]]] // Last; a[n_] := Select[counts, #[[2]] == n &] // Last // First; Table[a[n], {n, 0, 51}] (* Jean-Fran├žois Alcover, Jul 12 2012 *)

CROSSREFS

Cf. A080654, A025429.

Sequence in context: A039380 A043203 A043983 * A140156 A154600 A100593

Adjacent sequences:  A080670 A080671 A080672 * A080674 A080675 A080676

KEYWORD

easy,nice,nonn

AUTHOR

Rainer Rosenthal, Mar 02 2003

EXTENSIONS

a(15)-a(51) from Donovan Johnson, Aug 23 2010

Definition adjusted to cope with otherwise undefined values and b-file extended by Hagen von Eitzen, Jun 04 2014

STATUS

approved

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Last modified October 25 01:40 EDT 2021. Contains 348233 sequences. (Running on oeis4.)