

A360530


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


3



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, 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, 4, 3, 3, 3, 2, 2, 3, 3, 4, 3, 3, 2, 1, 2, 3, 4, 2, 3, 3
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OFFSET

1,2


COMMENTS

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


REFERENCES

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


LINKS



FORMULA

a(n) <= 4. Proof: With Lagrange's foursquare 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


EXAMPLE

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.


PROG

(PARI)
findsquare(k, m) = if(k == 1, issquare(m), for(j=1, m, if(j*j+k > m, return(0), if(findsquare(k1, mj*j), return(1)))));
a(n) = for(t = 1, n+1, if(findsquare(t, n*t), return(t)));


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



