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a(n) is the smallest positive integer k such that n can be expressed as the arithmetic mean of k nonzero squares.
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%I #92 May 02 2023 06:31:52

%S 1,3,3,1,2,3,3,3,1,2,3,3,2,3,3,1,2,3,3,2,4,3,3,3,1,2,3,3,2,3,3,3,3,2,

%T 3,1,2,3,3,2,2,3,3,3,2,3,3,3,1,2,3,2,2,3,3,3,3,2,3,3,2,3,3,1,2,3,3,2,

%U 4,3,3,3,2,2,3,3,4,3,3,2,1,2,3,4,2,3,3

%N a(n) is the smallest positive integer k such that n can be expressed as the arithmetic mean of k nonzero squares.

%C a(n) is the smallest number k such that n*k can be expressed as the sum of k nonzero squares.

%D J. H. Conway, The Sensual (Quadratic) Form, M.A.A., 1997, p. 140.

%H Yifan Xie, <a href="/A360530/b360530.txt">Table of n, a(n) for n = 1..10000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem">Lagrange's four-square theorem</a>

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>

%F a(n) <= 4. Proof: With Lagrange's four-square theorem, if 4*n is not the sum of 4 positive squares (see A000534), then it is easy to express 3*n as the sum of 3 positive squares. - _Yifan Xie_ and _Thomas Scheuerle_, Apr 29 2023

%e For n = 2, if k = 1, 2*1 = 2 is a nonsquare; if k = 2, 2*2 = 4 cannot be expressed as the sum of 2 nonzero squares; if k = 3, 2*3 = 6 = 2^2+1^2+1^2, so a(2) = 3.

%o (PARI)

%o findsquare(k, m) = if(k == 1, issquare(m), for(j=1, m, if(j*j+k > m, return(0), if(findsquare(k-1, m-j*j), return(1)))));

%o a(n) = for(t = 1, n+1, if(findsquare(t, n*t), return(t)));

%Y Cf. A000290, A000378, A000404, A000408, A000414, A000534, A047700.

%Y Cf. A362068 (allows zeros), A362110 (distinct).

%K nonn,easy

%O 1,2

%A _Yifan Xie_, Apr 05 2023