A025375 -id:A025375 - cp's OEIS frontend

A025428 Number of partitions of n into 4 nonzero squares.

0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 3, 0, 1, 2, 0, 1, 2, 1, 2, 2, 1, 2, 1, 0, 3, 2, 1, 2, 1, 2, 1, 2, 2, 1, 4, 1, 2, 3, 0, 2, 4, 1, 3, 2, 1, 4, 1, 1, 3, 3, 2, 2, 4, 2, 1, 3, 2, 3, 4, 2, 3, 3, 1, 2, 5, 2, 4, 3, 2, 4, 1, 1, 6, 4, 3, 4, 2, 3, 0, 4, 4, 3, 5, 1, 5, 5, 1, 4, 5, 2

Offset: 0

David W. Wilson

Records occur at n= 4, 28, 52, 82, 90, 130, 162, 198, 202, 210,.... - R. J. Mathar, Sep 15 2015

Cf. A000414, A000534, A025357-A025375, A216374, A025416 (greedy inverse).

Column k=4 of A243148.

A025428 := proc(n)
    local a,i,j,k,lsq ;
    a := 0 ;
    for i from 1 do
        if 4*i^2 > n then
            return a;
        end if;
        for j from i do
            if i^2+3*j^2 > n then
                break;
            end if;
            for k from j do
                if i^2+j^2+2*k^2 > n then
                    break;
                end if;
                lsq := n-i^2-j^2-k^2 ;
                if lsq >= k^2 and issqr(lsq) then
                    a := a+1 ;
                end if;
            end do:
        end do:
    end do:
end proc:
seq(A025428(n),n=1..40) ; # R. J. Mathar, Jun 15 2018
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
     `if`(i<1 or t<1, 0, b(n, i-1, t)+`if`(i^2>n, 0, b(n-i^2, i, t-1))))
    end:
a:= n-> b(n, isqrt(n), 4):
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 14 2019

nn = 100; lim = Sqrt[nn]; t = Table[0, {nn}]; Do[n = a^2 + b^2 + c^2 + d^2; If[n <= nn, t[[n]]++], {a, lim}, {b, a, lim}, {c, b, lim}, {d, c, lim}]; t (* T. D. Noe, Sep 28 2012 *)
f[n_] := Length@ IntegerPartitions[n, {4}, Range[ Floor[ Sqrt[n - 1]]]^2]; Array[f, 105] (* Robert G. Wilson v, Sep 28 2012 *)

A025428(n)=sum(a=1,n,sum(b=1,a,sum(c=1,b,sum(d=1,c,a^2+b^2+c^2+d^2==n))))

A025428(n)=sum(a=1,sqrtint(max(n-3,0)), sum(b=1,min(sqrtint(n-a^2-2),a), sum(c=1,min(sqrtint(n-a^2-b^2-1),b),issquare(n-a^2-b^2-c^2,&d) & d <= c )))

A025428(n)=sum(a=sqrtint(max(n,4)\4),sqrtint(max(n-3,0)), sum(b=sqrtint((n-a^2)\3-1)+1,min(sqrtint(n-a^2-2),a), sum(c=sqrtint((t=n-a^2-b^2)\2-1)+1, min(sqrtint(t-1),b), issquare(t-c^2) ))) \\ - M. F. Hasler, Sep 17 2012
for(n=1,100,print1(A025428(n),","))

T(n)={a=matrix(n,4,i,j,0);for(d=1,sqrtint(n),forstep(i=n,d*d+1,-1,for(j=2,4,a[i,j]+=sum(k=1,j,if(k0,a[i-k*d*d,j-k],if(k==j&&i-k*d*d==0,1)))));a[d*d,1]=1);for(i=1,n,print(i" "a[i,4]))} /* Robert Gerbicz, Sep 28 2012 */

For n>0, a(n) = ( A063730(n) + 6*A213024(n) + 3*A063725(n/2) + 8*A092573(n) + 6*A010052(n/4) ) / 24. - Max Alekseyev, Sep 30 2012
a(n) = ( A000118(n) - 4*A005875(n) - 6*A004018(n) - 12*A000122(n) - 15*A000007(n) + 12*A014455(n) - 24*A033715(n) - 12*A000122(n/2) + 12*A004018(n/2) + 32*A033716(n) - 32*A000122(n/3) + 48*A000122(n/4) ) / 384. - Max Alekseyev, Sep 30 2012
a(n) = [x^n y^4] Product_{k>=1} 1/(1 - y*x^(k^2)). - Ilya Gutkovskiy, Apr 19 2019
a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} A010052(i) * A010052(j) * A010052(k) * A010052(n-i-j-k). - Wesley Ivan Hurt, Apr 19 2019

Values of a(0..10^4) double-checked by M. F. Hasler, Sep 17 2012

A025338 Numbers that are the sum of 3 nonzero squares in 10 or more ways.

Original entry on oeis.org

594, 734, 761, 794, 801, 846, 854, 866, 881, 909, 926, 941, 950, 965, 986, 1001, 1026, 1034, 1041, 1046, 1049, 1089, 1106, 1109, 1121, 1130, 1154, 1161, 1169, 1181, 1190, 1206, 1209, 1214, 1226, 1238, 1265, 1274, 1286, 1301, 1314, 1322, 1326, 1329, 1341

Offset: 1

David W. Wilson

nonn

Robert Price, Table of n, a(n) for n = 1..4009
Index entries for sequences related to sums of squares

Cf. A024796, A025301, A025322, A025323, A025324, A025325, A025326, A025327, A025328, A025329, A025330, A025331, A025332, A025333, A025334, A025335, A025336, A025337, A025338, A025375, A025427, A345121.

A345155 Numbers that are the sum of four third powers in ten or more ways.

Original entry on oeis.org

21896, 36225, 46872, 48321, 48825, 51506, 52416, 53200, 55575, 58338, 58968, 59059, 60480, 62244, 66024, 67536, 67851, 70434, 70525, 71155, 72819, 73808, 76384, 76923, 77896, 78624, 78912, 81081, 81991, 85995, 87507, 88641, 90181, 90783, 91448, 91728, 92008

Offset: 1

David Consiglio, Jr., Jun 09 2021

nonn

			21896 is a term because 21896 = 1^3 + 11^3 + 19^3 + 22^3  = 2^3 + 2^3 + 12^3 + 26^3  = 2^3 + 3^3 + 19^3 + 23^3  = 2^3 + 5^3 + 15^3 + 25^3  = 3^3 + 10^3 + 16^3 + 24^3  = 3^3 + 17^3 + 19^3 + 19^3  = 4^3 + 6^3 + 20^3 + 22^3  = 5^3 + 8^3 + 14^3 + 25^3  = 7^3 + 11^3 + 17^3 + 23^3  = 8^3 + 9^3 + 19^3 + 22^3.

David Consiglio, Jr., Table of n, a(n) for n = 1..10000

Cf. A025375, A344928, A345121, A345146, A345156, A345187.

from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**3 for x in range(1, 1000)]
for pos in cwr(power_terms, 4):
    tot = sum(pos)
    keep[tot] += 1
rets = sorted([k for k, v in keep.items() if v >= 10])
for x in range(len(rets)):
    print(rets[x])

A025374 Numbers that are the sum of 4 nonzero squares in 9 or more ways.

Original entry on oeis.org

162, 178, 198, 202, 207, 210, 220, 223, 225, 226, 231, 234, 242, 243, 246, 247, 250, 252, 253, 255, 258, 262, 265, 266, 267, 268, 270, 271, 273, 274, 278, 279, 282, 283, 285, 286, 287, 290, 291, 292, 294, 295, 297, 298, 300, 301, 303, 306, 307, 309, 310, 313, 314, 315

Offset: 1

David W. Wilson

nonn

Index entries for sequences related to sums of squares

Cf. A025337, A025365, A025373, A025375, A344802, A345146.

{n: A025428(n) >= 9}. - R. J. Mathar, Jun 15 2018

A344803 Numbers that are the sum of five squares in ten or more ways.

Original entry on oeis.org

107, 109, 116, 125, 128, 131, 133, 134, 136, 139, 140, 142, 146, 147, 148, 149, 151, 152, 154, 155, 157, 158, 160, 163, 164, 166, 167, 168, 170, 171, 172, 173, 174, 175, 176, 178, 179, 181, 182, 183, 184, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196

Offset: 1

Sean A. Irvine, May 29 2021

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..10000

Cf. A025375, A294744, A344802, A345187, A345477.

A025394 Numbers that are the sum of 4 distinct nonzero squares in 10 or more ways.

Original entry on oeis.org

270, 294, 318, 330, 342, 350, 351, 354, 366, 375, 378, 382, 390, 398, 399, 402, 406, 410, 414, 422, 426, 429, 430, 434, 435, 438, 441, 442, 446, 450, 455, 459, 462, 465, 466, 470, 471, 474, 478, 483, 486, 490, 494, 495, 498, 501, 506, 507, 510, 513, 515, 518, 519, 522

Offset: 1

David W. Wilson

nonn

The common sum of rows, columns and diagonals of magic squares whose entries are distinct squares must be terms of this sequence. - M. F. Hasler, Jul 22 2025

M. F. Hasler, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of squares

Cf. A025443 (number of ways to write n as sum of 4 distinct nonzero squares).
Cf. A025385 (subsequence of terms with A025443(n) = 10 exactly),
Subsequence of A025375 (sums of 4 not necessarily distinct nonzero squares in at least 10 ways).

{n: A025443(n) >= 10}. - R. J. Mathar, Jun 15 2018

A024375 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023532, t = A023533.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A023532, A023533.

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
A023532:= func< n |  IsSquare(8*n+9) select 0 else 1 >;
A025375:= func< n | (&+[A023532(k)*A023533(n+1-k): k in [1..Floor((n+1)/2)]]) >;
[A025375(n): n in [1..130]]; // G. C. Greubel, Sep 07 2022

A023533[n_]:= A023533[n]= If[Binomial[Floor[Surd[6*n-1, 3]] +2, 3] != n, 0, 1];
A023532[n_]:= If[IntegerQ[(Sqrt[8*n+9] -3)/2], 0, 1];
A025375[n_]:= A025075[n]= Sum[A023532[j]*A023533[n-j+1], {j, Floor[(n+1)/2]}];
Table[A025375[n], {n, 130}] (* G. C. Greubel, Sep 07 2022 *)

@CachedFunction
def A023533(n): return 0 if (binomial( floor( (6*n-1)^(1/3) ) +2, 3) != n) else 1
def A023532(n): return 0 if is_square(8*n+9) else 1
def A025375(n): return sum(A023532(k)*A023533(n-k+1) for k in (1..((n+1)//2)))
[A025375(n) for n in (1..130)] # G. C. Greubel, Sep 07 2022

cp's OEIS Frontend

A025428 Number of partitions of n into 4 nonzero squares.

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A025338 Numbers that are the sum of 3 nonzero squares in 10 or more ways.

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A345155 Numbers that are the sum of four third powers in ten or more ways.

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A025374 Numbers that are the sum of 4 nonzero squares in 9 or more ways.

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A344803 Numbers that are the sum of five squares in ten or more ways.

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A025394 Numbers that are the sum of 4 distinct nonzero squares in 10 or more ways.

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A024375 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023532, t = A023533.

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A024375 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023532, t = A023533.