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

1,2

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.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Index entries for sequences related to sums of squares

Example

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.

Maple

b:= proc(n, i, t) option remember; `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)))))
    end:
a:= proc(n) local k;
      for k while b(k, isqrt(k), n)<n do od; k
    end:
seq(a(n), n=1..100);

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 *)

Crossrefs

Cf. A052261.

Sequence in context: A294297 A320673 A081646 * A052261 A295155 A118146

Adjacent sequences: A215905 A215906 A215907 * A215909 A215910 A215911

Keyword

nonn

Author

Alois P. Heinz, Aug 26 2012

Status

approved