

A004437


Numbers that are not the sum of 4 distinct squares.


1



1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 27, 28, 31, 32, 33, 34, 36, 37, 40, 43, 44, 47, 48, 52, 55, 58, 60, 64, 67, 68, 72, 73, 76, 80, 82, 88, 92, 96, 97, 100, 103, 108, 112
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OFFSET

1,2


COMMENTS

It follows from the formula that there are infinitely many integers that cannot be partitioned into a sum of four distinct squares of nonnegative integers and infinitely many that can. Furthermore, the largest odd number that has no such partition is 103, and thereafter the terms satisfy the thirtyfirst order recurrence relation a(n) = 4a(n31).  Ant King, Nov 02 2010


LINKS

Table of n, a(n) for n=1..56.
Gordon Pall, On Sums of Squares, The American Mathematical Monthly, Vol. 40, No. 1, (January 1933), pp. 1018. [From Ant King, Nov 02 2010]
Index entries for sequences related to sums of squares


FORMULA

Let k>=0. Then the only integers that cannot be partitioned into a sum of four distinct squares of nonnegative integers are 4^k * N3, where N3 = (N1 union N2), and N1 and N2 are defined by N1 = {1,3,5,7,9,11,13,15,17,19,23,25,27,31,33,37,43,47,55,67,73,97,103} and N2 = {2,6,10,18,22,34,58,82}, respectively.  Ant King, Nov 02 2010


MATHEMATICA

data = Reduce[ w^2 + x^2 + y^2 + z^2 == # && 0 <= w < x < y < z < #, {w, x, y, z}, Integers] & /@ Range[112]; DeleteCases[ Table[If[Head[data[[k]]] === Symbol, k, 0], {k, 1, Length[data]}], 0] (* Ant King, Nov 02 2010 *)


CROSSREFS

Cf. A001944 (complement).
Sequence in context: A022772 A004440 A026495 * A298534 A291442 A236866
Adjacent sequences: A004434 A004435 A004436 * A004438 A004439 A004440


KEYWORD

nonn


AUTHOR

N. J. A. Sloane


STATUS

approved



