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

%I A215908 #33 Mar 23 2023 16:58:45
%S A215908 1,50,54,52,53,54,55,56,57,61,65,66,67,68,74,78,79,81,82,83,84,88,92,
%T A215908 93,96,98,99,100,101,102,106,107,108,112,113,114,115,116,117,121,124,
%U A215908 125,129,130,131,132,133,134,136,137,141,142,143,147,148,149,150,151
%N A215908 Smallest integer that can be represented as the sum of n squares of positive integers in at least n distinct ways.
%C A215908 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.
%H A215908 Alois P. Heinz, <a href="/A215908/b215908.txt">Table of n, a(n) for n = 1..5000</a>
%H A215908 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%e A215908 a(1) =  1 = 1^2.
%e A215908 a(2) = 50 = 1^2+7^2 = 5^2+5^2.
%e A215908 a(3) = 54 = 2^2+2*5^2 = 2*3^2+6^2 = 1^2+2^2+7^2.
%e A215908 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.
%p A215908 b:= proc(n, i, t) option remember; `if`(n<t, 0, `if`(n=t, 1,
%p A215908       `if`(t=0, 0, `if`(i>0, b(n, i-1, t), 0)+
%p A215908       `if`(i^2>n, 0, b(n-i^2, i, t-1)))))
%p A215908     end:
%p A215908 a:= proc(n) local k;
%p A215908       for k while b(k, isqrt(k), n)<n do od; k
%p A215908     end:
%p A215908 seq(a(n), n=1..100);
%t A215908 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 *)
%Y A215908 Cf. A052261.
%K A215908 nonn
%O A215908 1,2
%A A215908 _Alois P. Heinz_, Aug 26 2012