A025376 -id:A025376

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A025443

Number of partitions of n into 4 distinct nonzero squares.

+10
32

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,79

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Index entries for sequences related to sums of squares

FORMULA

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

MAPLE

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),

`if`(t*i^2<n, 0, `if`(i=1, 0, b(n, i-1, t))+

`if`(i^2>n, 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

MATHEMATICA

b[n_, i_, t_] := b[n, i, t] = If[n==0, If[t==0, 1, 0], If[t*i^2<n, 0, If[i == 1, 0, b[n, i-1, t]] + If[i^2>n, 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 *)

CROSSREFS

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

Column k=4 of A341040.

KEYWORD

nonn,look

AUTHOR

David W. Wilson

STATUS

approved

A004433

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

+10
17

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

Index entries for sequences related to sums of squares

FORMULA

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

EXAMPLE

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

MATHEMATICA

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 *)

PROG

(Haskell)

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

(PARI) list(lim)=my(v=List()); lim\=1; for(z=4, sqrtint(lim\4), for(y=3, min(sqrtint((lim-z^2)\3), z-1), for(x=2, min(sqrtint((lim-y^2-z^2)\2), y-1), for(w=1, min(sqrtint(lim-x^2-y^2-z^2), x-1), listput(v, w^2+x^2+y^2+z^2))))); Set(v) \\ Charles R Greathouse IV, Feb 07 2017

CROSSREFS

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

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

A025386

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..55.

Index entries for sequences related to sums of squares

FORMULA

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

KEYWORD

nonn

AUTHOR

David W. Wilson

STATUS

approved

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