A001944 -id:A001944 - cp's OEIS frontend

A004433 Numbers that are the sum of 4 distinct nonzero squares: of form w^2+x^2+y^2+z^2 with 0

30, 39, 46, 50, 51, 54, 57, 62, 63, 65, 66, 70, 71, 74, 75, 78, 79, 81, 84, 85, 86, 87, 90, 91, 93, 94, 95, 98, 99, 102, 105, 106, 107, 109, 110, 111, 113, 114, 116, 117, 118, 119, 120, 121, 122, 123, 125, 126, 127, 129, 130, 131, 133, 134, 135, 137

Offset: 1

N. J. A. Sloane

			30 = 1^2+2^2+3^2+4^2.

Cf. A001944, A001995, A003995, A004431, A004432, A004434, A224981, A224982, A224983, A000290.

a004433 n = a004433_list !! (n-1)
a004433_list = filter (p 4 $ tail a000290_list) [1..] where
   p k (q:qs) m = k == 0 && m == 0 ||
                  q <= m && k >= 0 && (p (k - 1) qs (m - q) || p k qs m)
-- Reinhard Zumkeller, Apr 22 2013

data = Flatten[ DeleteCases[ FindInstance[ w^2 + x^2 + y^2 + z^2 == # && 0 < w < x < y < z < #, {w, x, y, z}, Integers] & /@ Range[137], {}], 1]; w^2 + x^2 + y^2 + z^2 /. data (* Ant King, Oct 17 2010 *)
Select[Union[Total[#^2]&/@Subsets[Range[10],{4}]],#<=137&] (* Harvey P. Dale, Jul 03 2011 *)

list(lim)=my(v=List([30, 39, 46, 50, 51, 54, 57, 62, 63, 65, 66, 70, 71, 74, 75, 78, 79, 81, 84, 85, 86, 87, 90, 91, 93, 94, 95, 98, 99, 102, 105, 106, 107, 109, 110, 111, 113, 114, 116, 117, 118, 119, 120, 121, 122, 123, 125, 126, 127, 129, 130, 131, 133, 134, 135, 137, 138, 139, 140, 141, 142, 143, 145, 146, 147, 149, 150, 151, 153, 154, 155, 156]), u=[160, 168, 172, 176, 188, 192, 208, 220, 224, 232, 240, 256, 268, 272, 288, 292, 304, 320, 328, 352, 368, 384, 388, 400, 412, 416, 432, 448, 496, 512, 528, 544, 576, 592, 608], t=1); if(lim<156, return(select(k->k<=lim, Vec(v)))); for(n=158,lim\1, if(n#u, t=1)); Vec(v) \\ Charles R Greathouse IV, Jan 08 2025

{n: A025443(n) >=1}. Union of A025386 and A025376. - R. J. Mathar, Jun 15 2018

a(n) = n + O(log n). - Charles R Greathouse IV, Jan 08 2025

A001995 Numbers that are the sum of 5 distinct squares: of form v^2 + w^2 + x^2 + y^2 + z^2 with 0 <= v < w < x < y < z.

Original entry on oeis.org

30, 39, 46, 50, 51, 54, 55, 57, 62, 63, 65, 66, 70, 71, 74, 75, 78, 79, 81, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 98, 99, 100, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 125, 126, 127, 129, 130, 131

Offset: 1

N. J. A. Sloane

nonn

			30 = 0^2 + 1^1 + 2^2 + 3^2 + 4^2.

Cf. A001944.

nn = 15; Select[Union[Flatten[Table[a^2 + b^2 + c^2 + d^2 + e^2, {a, 0, nn}, {b, a + 1, nn}, {c, b + 1, nn}, {d, c + 1, nn}, {e, d + 1, nn}]]], # <= nn^2 &] (* T. D. Noe, Aug 17 2012 *)

A001948 These numbers when multiplied by all powers of 4 give the numbers that are not the sums of 4 distinct squares.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 15, 17, 18, 19, 22, 23, 25, 27, 31, 33, 34, 37, 43, 47, 55, 58, 67, 73, 82, 97, 103

Offset: 1

N. J. A. Sloane, Dan Hoey

See also the comment in sequence A004437: The only integers that cannot be partitioned into a sum of four distinct squares of nonnegative integers are 4^k x (A union B) where A = {1,3,5,7,9,11,13,15,17,19,23,25,27,31,33,37,43,47,55,67,73,97,103} and B = {2,6,10,18,22,34,58,82}. - M. F. Hasler, Jun 11 2014

Gordon Pall, On Sums of Squares, The American Mathematical Monthly, Vol. 40, No. 1, (January 1933), pp. 10-18.
Index entries for sequences related to sums of squares

Cf. A001944, A004437.

A004437 Numbers that are not the sum of 4 distinct squares.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

nonn

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). - Ant King, Nov 02 2010

Gordon Pall, On Sums of Squares, The American Mathematical Monthly, Vol. 40, No. 1, (January 1933), pp. 10-18. [From _Ant King_, Nov 02 2010]
Index entries for sequences related to sums of squares

Cf. A001944 (complement).

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 *)

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

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A004433 Numbers that are the sum of 4 distinct nonzero squares: of form w^2+x^2+y^2+z^2 with 0

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A001995 Numbers that are the sum of 5 distinct squares: of form v^2 + w^2 + x^2 + y^2 + z^2 with 0 <= v < w < x < y < z.

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A001948 These numbers when multiplied by all powers of 4 give the numbers that are not the sums of 4 distinct squares.

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A004437 Numbers that are not the sum of 4 distinct squares.

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