A025376 -id:A025376 - cp's OEIS frontend

A025443 Number of partitions of n into 4 distinct nonzero squares.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 3, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 2, 1, 0, 1, 2, 2, 0, 0, 1, 2, 0, 0, 3, 0, 0, 2, 1, 1

Offset: 0

David W. Wilson

Alois P. Heinz, Table of n, a(n) for n = 0..20000
Index entries for sequences related to sums of squares

Cf. A025428 (not necessarily distinct), A025376-A025394 (subsequences), A025417 (greedy inverse).

Column k=4 of A341040.

b:= proc(n,i,t) option remember; `if`(n=0, `if`(t=0,1,0),
      `if`(t*i^2n, 0, b(n-i^2,i-1,t-1))))
    end:
a:= n-> b(n, isqrt(n), 4):
seq(a(n), n=0..150);  # Alois P. Heinz, Feb 07 2013

b[n_, i_, t_] := b[n, i, t] = If[n==0, If[t==0, 1, 0], If[t*i^2n, 0, b[n-i^2, i-1, t-1]]]]; a[n_] := b[n, Sqrt[n] // Floor, 4]; Table[a[n], {n, 0, 150}] (* Jean-François Alcover, Feb 29 2016, after Alois P. Heinz*)
dnzs[n_]:=Length[Select[IntegerPartitions[n,{4}],Length[Union[#]]==4&&AllTrue[ Sqrt[ #], IntegerQ] && FreeQ[#,0]&]]; Array[dnzs,110,0] (* Harvey P. Dale, Jun 09 2024 *)

a(n) = [x^n y^4] Product_{k>=1} (1 + y*x^(k^2)). - Ilya Gutkovskiy, Apr 22 2019

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

Original entry on oeis.org

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

A025386 Numbers that are the sum of 4 distinct nonzero squares in 2 or more ways.

Original entry on oeis.org

78, 90, 94, 95, 99, 102, 105, 110, 111, 114, 119, 123, 126, 129, 130, 134, 135, 138, 141, 142, 143, 146, 147, 150, 151, 153, 154, 155, 156, 158, 159, 162, 165, 166, 167, 169, 170, 171, 174, 175, 177, 179, 182, 183, 185, 186, 189, 190, 191, 193, 194, 195, 197, 198, 199

Offset: 1

David W. Wilson

nonn

Index entries for sequences related to sums of squares

{n: A025443(n) >= 2}. Union of A025387 and A025376. - R. J. Mathar, Jun 15 2018

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A025443 Number of partitions of n into 4 distinct nonzero squares.

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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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A025386 Numbers that are the sum of 4 distinct nonzero squares in 2 or more ways.

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