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A004437
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Numbers that are not the sum of 4 distinct squares.
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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 thirty first order recurrence relation a(n)=4a(n-31). [From Ant KIng (mathstutoring(AT)ntlworld.com), Nov 02 2010]
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REFERENCES
| Pall, Gordon; On Sums of Squares, The American Mathematical Monthly, Vol. 40, No. 1, (January 1933), pp. 10-18. [From Ant KIng (mathstutoring(AT)ntlworld.com), Nov 02 2010]
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LINKS
| Index entries for sequences related to sums of squares
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FORMULA
| Let k>=0. Then the only integers that cannot be partitioned into a sum of four distinct squares of non-negative integers are 4^k x 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. [From Ant KIng (mathstutoring(AT)ntlworld.com), Nov 02 2010]
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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] [From Ant KIng (mathstutoring(AT)ntlworld.com), Nov 02 2010]
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CROSSREFS
| Sequence in context: A022772 A004440 A026495 * A122526 A120401 A127034
Adjacent sequences: A004434 A004435 A004436 * A004438 A004439 A004440
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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