login
Numbers that are a sum of four nonzero squares where the summands have no common square factor > 1.
4

%I #14 Dec 12 2015 21:37:30

%S 4,7,10,12,13,15,18,19,20,21,22,23,25,26,27,28,30,31,33,34,35,36,37,

%T 38,39,42,43,44,45,46,47,49,50,51,52,53,54,55,57,58,59,60,61,62,63,65,

%U 66,67,68,69,70,71,73,74,75,76,77,78,79,81,82,83,84,85,86,87,89,90,91,92,93,94,95,97,98,99,100,101,102

%N Numbers that are a sum of four nonzero squares where the summands have no common square factor > 1.

%C A representation of a(n) as a sum of four nonzero squares is denoted by [s(1),s(2),s(3),s(4)] with nondecreasing entries > 1 and Sum_{j=1..4} s(j)^2 = a(n). It is called primitive if gcd(s(1),s(2),s(3),s(4)) = 1. a(n) is the number with at least one such primitive representation for n, and the multiplicity m is given by the non-vanishing entries of A097203, that is A097203(a(n)).

%F A097203(a(n)) is not 0.

%e a(1) = 4 because 4 has the primitive representation [1, 1, 1, 1].

%e a(16) = 28, because 28 has the primitive representations [1, 1, 1, 5] and [1, 3, 3, 3] ([2, 2, 2, 4] is not primitive.).

%e 4: [1, 1, 1, 1], 7: [1, 1, 1, 2], 10: [1, 1, 2, 2], 12: [1, 1, 1, 3], 13: [1, 2, 2, 2], 15: [1, 1, 2, 3], 18: [1, 2, 2, 3],

%e 19: [1, 1, 1, 4], 20: [1, 1, 3, 3], 21: [2, 2, 2, 3], 22: [1, 1, 2, 4], 23: [1, 2, 3, 3], 25: [1, 2, 2, 4], 26: [2, 2, 3, 3], 27: [1, 1, 3, 4], 28: [1, 1, 1, 5], [1, 3, 3, 3], ...

%Y Cf. A097203.

%K nonn

%O 1,1

%A _Wolfdieter Lang_, Mar 25 2013