A080673 Largest number with exactly n representations as sum of five positive squares (or 0 if no number with exactly n representations exists).

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

Offset: 0

Author

Rainer Rosenthal, Mar 02 2003

Keywords

• easy
• nice
• nonn

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

Examples

			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.
		

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

Crossrefs

Cf. A080654, A025429.

Programs

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-François Alcover, Jul 12 2012 *)

Formula

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

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