A120951 -id:A120951 - cp's OEIS frontend

A129210 Largest number not the sum of n distinct nonzero squares.

245, 333, 330, 462, 539, 647, 888, 1036, 1177, 1445, 1722, 1990, 2311, 2672, 3047, 3492, 4093, 4613, 5138, 5718, 6379, 7123, 7952, 8676, 9537, 10393, 11558, 12602, 13743, 14863, 16252, 17528, 18957, 20481, 22042, 23678, 25347, 27207, 29092

Offset: 5

T. D. Noe, Apr 03 2007

nonn

Halter-Koch essentially finds (5)-a(12) (with a coprimality condition, but Bateman, Hildebrand, & Purdy show that this can be dropped). - Charles R Greathouse IV, Mar 18 2014

T. D. Noe, Table of n, a(n) for n = 5..400 (from Bateman et al.)
Paul T. Bateman, Adolf J. Hildebrand, and George B. Purdy, Sums of distinct squares, Acta Arithmetica 67 (1994), pp. 349-380.
Franz Halter-Koch, Darstellung natürlicher Zahlen als Summe von Quadraten, Acta Arithmetica 42 (1982), pp. 11-20.

Cf. A120951 (numbers that are not the sum of 5 distinct nonzero squares).

Bateman, Hildebrand, & Purdy prove that a(n) = n^3/3 + n^2/2 + sqrt(8)*n^(3/2) + O(n^(5/4)), see their Theorem 1. - Charles R Greathouse IV, Mar 31 2025

A004438 Numbers that are not the sum of 5 distinct squares.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 47, 48, 49, 52, 53, 56, 58, 59, 60, 61, 64, 67, 68, 69, 72, 73, 76, 77, 80, 83, 89, 92, 96, 97, 101, 104, 108, 112, 124, 128, 136, 188, 224

Offset: 1

N. J. A. Sloane

Paul T. Bateman, Adolf J. Hildebrand and George B. Purdy, Sums of distinct squares, Acta Arith. 67 (1994), 349-380.
E. M. Wright, The representation of a number as a sum of five or more squares, Quart. J. Math. Oxford Ser. 4 (1933), 37-51.
Index entries for sequences related to sums of squares

Cf. A120951.

nn=10000; t=Table[0,{nn}]; lim=Floor[Sqrt[nn]]; sq=Table[i^2, {i,0,lim}]; lim=lim+1; Do[n=sq[[i1]]+sq[[i2]]+sq[[i3]]+sq[[i4]]+sq[[i5]]; If[n<=nn, t[[n]]++ ], {i1,lim}, {i2,i1+1,lim}, {i3,i2+1,lim}, {i4,i3+1,lim}, {i5,i4+1,lim}]; Flatten[Position[t,0]] (* T. D. Noe, Jul 24 2006 *)
j[{a_, b_, c_, d_, e_}] := If[0 <= a < b < c < d < e, a^2 + b^2 + c^2 + d^2 + e^2, False]; Union[ Complement[Range[500], DeleteCases[ Flatten[Map[j, PowersRepresentations[ #, 5, 2]] & /@ Range[500]], False]]] (* Ant King, Sep 30 2010 *)

More terms from Franklin T. Adams-Watters, Jul 18 2006

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A129210 Largest number not the sum of n distinct nonzero squares.

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A004438 Numbers that are not the sum of 5 distinct squares.

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