A215908 Smallest integer that can be represented as the sum of n squares of positive integers in at least n distinct ways.

1, 50, 54, 52, 53, 54, 55, 56, 57, 61, 65, 66, 67, 68, 74, 78, 79, 81, 82, 83, 84, 88, 92, 93, 96, 98, 99, 100, 101, 102, 106, 107, 108, 112, 113, 114, 115, 116, 117, 121, 124, 125, 129, 130, 131, 132, 133, 134, 136, 137, 141, 142, 143, 147, 148, 149, 150, 151

Offset: 1

Author

Alois P. Heinz, Aug 26 2012

Keywords

• nonn

Comments

This sequence differs from A052261 first at n=11: a(11) = 65 < A052261(11) = 67. 65 has 12 distinct representations (as the sum of 11 squares of positive integers) whereas 67 has exactly 11.

Examples

			a(1) =  1 = 1^2.
a(2) = 50 = 1^2+7^2 = 5^2+5^2.
a(3) = 54 = 2^2+2*5^2 = 2*3^2+6^2 = 1^2+2^2+7^2.
a(11) = 65 = 3*1^2+2*2^2+6*3^2 = 2*1^2+5*2^2+3*3^2+4^2 = 1^2+8*2^2+2*4^2 = 6*1^2+3*3^2+2*4^2 = 5*1^2+3*2^2+3*4^2 = 10*2^2+5^2 = 5*1^2+2*2^2+3*3^2+5^2 = 4*1^2+5*2^2+4^2+5^2 = 8*1^2+2*4^2+5^2 = 7*1^2+2*2^2+2*5^2 = 7*1^2+2^2+2*3^2+6^2 = 8*1^2+2*2^2+7^2.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..5000
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A052261.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n0, b(n, i-1, t), 0)+
      `if`(i^2>n, 0, b(n-i^2, i, t-1)))))
    end:
a:= proc(n) local k;
      for k while b(k, isqrt(k), n)
				

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n < t, 0, If[n == t, 1, If[t == 0, 0, If[i > 0, b[n, i-1, t], 0] + If[i^2 > n, 0, b[n-i^2, i, t-1]]]]]; a[n_] := Module[{k}, For[k = 1, b[k, Sqrt[k] // Floor, n] < n, k++]; k]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 30 2013, translated from Maple *)