A002635 -id:A002635 - cp's OEIS frontend

A297788 Number of partitions of n into 3 squares and a nonnegative cube.

1, 2, 2, 2, 2, 2, 2, 1, 2, 4, 4, 3, 3, 3, 3, 1, 2, 5, 5, 4, 3, 3, 3, 1, 2, 5, 6, 6, 4, 4, 5, 2, 3, 6, 6, 6, 5, 6, 5, 3, 3, 7, 6, 4, 6, 6, 6, 2, 3, 7, 6, 7, 6, 7, 8, 3, 4, 6, 6, 6, 5, 6, 8, 4, 4, 9, 8, 8, 7, 8, 7, 2, 6, 10, 9, 8, 8, 9, 7, 2, 6, 12, 11, 8, 7, 7

Offset: 0

XU Pingya, Jan 06 2018

nonn

When n is not of the form 4^a * (8b + 7), according to Legendre's three-square theorem, n = x^2 + y^2 + z^2 = x^2 + y^2 + z^2 + 0^3 (where a, b, x, y and z are nonnegative integers with x <= y <= z).
If n = 8b + 7, then n - 1 = 8b + 6 is not of the form 4^a * (8b + 7). So n = (n - 1) + 1 = x^2 + y^2 + z^2 + 1^3.
If n = 4 * (8b + 7), then n - 1 = 8 * (4b + 3) + 3 is also not of the form 4^a * (8b + 7).
If n = 4^2 * (8b + 7), then n - 8 = 4 * (8 * (4b + 3) + 2) is not of the form 4^a * (8b + 7).  n = (n - 8) + 8 = x^2 + y^2 + z^2 + 2^3.
If n = 4^k * (8b + 7) (k >= 3), then n - 8 = 4 * (8 * (4^(k - 1) * b + 4^(k - 3) * 14) - 2) = 4 * (8m - 2) is also not of the form 4^a * (8b + 7).
That is, every nonnegative integer can be represented as the sum of 3 squares and a nonnegative cube, so a(n) > 0.

			2 = 0^2 + 0^2 + 1^2 + 1^3 = 0^2 + 1^2 + 0^2 + 1^3, a(2) = 2.
9 = 0^2 + 0^2 + 1^2 + 2^3 = 0^2 + 1^2 + 0^2 + 2^3 = 0^2 + 2^2 + 2^2 + 1^3 = 1^2 + 2^2 + 2^2 + 0^3, a(9) = 4.

Robert Israel, Table of n, a(n) for n = 0..10000
Wikipedia, Legendre's three-square theorem

Cf. A002635, A004215, A274274.

N:= 100: # to get a(0)..a(N)
A:= Array(0..N):
for x from 0 to floor(sqrt(N)) do
  for y from 0 to x while x^2 + y^2 <= N do
    for z from 0 to y while x^2 + y^2 + z^2 <= N do
      for w from 0 do
        t:= x^2 + y^2 + z^2 + w^3;
        if t > N then break fi;
        A[t]:= A[t]+1;
od od od od:
convert(A,list); # Robert Israel, Jan 11 2018

a[n_]:=Sum[If[x^2+y^2+z^2+w^3==n, 1, 0], {x,0,n^(1/2)}, {y,x,(n-x^2)^(1/2)}, {z,y,(n-x^2-y^2)^(1/2)}, {w,0,(n-x^2-y^2-z^2)^(1/3)}]
Table[a[n], {n,0,86}]

A307531 a(n) is the greatest sum i + j + k + l where i^2 + j^2 + k^2 + l^2 = n and 0 <= i <= j <= k <= l.

Original entry on oeis.org

0, 1, 2, 3, 4, 3, 4, 5, 4, 5, 6, 5, 6, 7, 6, 7, 8, 7, 8, 7, 8, 9, 8, 9, 8, 9, 10, 9, 10, 9, 10, 11, 8, 11, 10, 11, 12, 11, 12, 11, 12, 11, 12, 13, 12, 13, 12, 13, 12, 13, 14, 13, 14, 13, 14, 13, 12, 15, 14, 15, 14, 15, 14, 15, 16, 15, 16, 15, 16, 15, 16, 15

Offset: 0

Rémy Sigrist, Apr 13 2019

nonn

The sequence is well defined as every nonnegative integer can be represented as a sum of four squares in at least one way.

It appears that a(n^2) = 2*n if n is even and 2*n-1 if n is odd. - Robert Israel, Apr 14 2019

			For n = 34:
- 34 can be expressed in 4 ways as a sum of four squares:
    i^2 + j^2 + k^2 + l^2   i+j+k+l
    ---------------------   -------
    0^2 + 0^2 + 3^2 + 5^2         8
    0^2 + 3^2 + 3^2 + 4^2        10
    1^2 + 1^2 + 4^2 + 4^2        10
    1^2 + 2^2 + 2^2 + 5^2        10
- a(34) = max(8, 10) = 10.

Robert Israel, Table of n, a(n) for n = 0..10000
Rémy Sigrist, C program for A307531
Wikipedia, Lagrange's four-square theorem

See A307510 for the multiplicative variant.

Cf. A002635, A354777.

C
```
See Links section.
```

g:= proc(n,k) option remember; local a;
  if k = 1 then if issqr(n) then return sqrt(n) else return -infinity fi fi;
  max(seq(a+procname(n-a^2,k-1),a=0..floor(sqrt(n))))
end proc:
seq(g(n,4), n=0..100); # Robert Israel, Apr 14 2019

Array[Max[Total /@ PowersRepresentations[#, 4, 2]] &, 68, 0] (* Michael De Vlieger, Apr 13 2019 *)

A161148 Number of partitions of n such that each term of the partition is a squared divisor of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 1, 5, 1, 4, 2, 6, 1, 9, 1, 8, 3, 6, 1, 16, 2, 7, 4, 12, 1, 21, 1, 15, 4, 9, 2, 39, 1, 10, 5, 25, 1, 35, 1, 24, 9, 12, 1, 76, 2, 21, 6, 32, 1, 61, 3, 38, 7, 15, 1, 174, 1, 16, 10, 46, 3, 93, 1, 50, 8, 42, 1, 231, 1, 19, 19, 60, 2, 135, 1

Offset: 1

R. J. Mathar, Jun 03 2009

nonn

			a(n=12)=5 counts these 5 partitions of 12: 1^2+1^2+..+1^2 = 1^2+1^2+...+1^2+2^2 = 1^2+1^2+..+1^2+2^2+2^2 = 1^2+1^2+1^2+3^2=2^2+2^2+2^2. Partitions with the divisors 4, 6 or 12 do not contribute to the count because 4^2, 6^2 and 12^2 are larger than n.

Cf. A000040, A130291, A018818, A001156, A002635.

a := proc(n) coeftayl(1/mul(1-x^(d^2),d=numtheory[divisors](n)),x=0,n) ; end:

a[n_] := SeriesCoefficient[1/Product[1-x^(d^2), {d, Divisors[n]}], {x, 0, n}];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Apr 04 2024, after Maple code *)

a(p) = 1 if p a prime (A000040).
a(2p) = A130291(n) if p=A000040(n).
a(n) = [x^n] Product_{d|n} 1/( 1-x^(d^2) ).

A293175 Integers with precisely six partitions into sums of four squares of nonnegative numbers.

Original entry on oeis.org

66, 81, 97, 99, 105, 110, 115, 121, 123, 124, 137, 139, 141, 149, 155, 156, 158, 159, 164, 179, 188, 239, 264, 284, 440, 496, 624, 632, 656, 752, 1056, 1136, 1760, 1984, 2496, 2528, 2624, 3008, 4224, 4544, 7040, 7936, 9984, 10112, 10496, 12032, 16896, 18176

Offset: 1

Robert Price, Oct 27 2017

nonn

A002635(a(n)) = 6.

D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.
Index entries for sequences related to sums of squares

Cf. A001156, A002635, A006431, A025428, A180149, A245022, A294282, A294297.

f[n_] := Length@ PowersRepresentations[n, 4, 2]; Select[ Range@ 19000, f@# == 6 &] (* Robert G. Wilson v, Oct 27 2017 *)

A294308 Integers with precisely seven partitions into sums of four squares of nonnegative numbers.

Original entry on oeis.org

82, 98, 100, 102, 106, 108, 118, 125, 129, 132, 133, 134, 135, 161, 163, 173, 183, 197, 199, 204, 211, 212, 215, 236, 263, 328, 392, 400, 408, 424, 432, 472, 528, 536, 816, 848, 944, 1312, 1568, 1600, 1632, 1696, 1728, 1888, 2112, 2144, 3264, 3392, 3776

Offset: 1

Robert Price, Oct 27 2017

nonn

A002635(a(n)) = 7.

Robert Price, Table of n, a(n) for n = 1..74
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.
Index entries for sequences related to sums of squares

Cf. A001156, A002635, A006431, A025428, A180149, A245022, A294282, A294297, A293175.

f[n_]:=Length@PowersRepresentations[n, 4, 2]; Select[Range@650, f@#==7 &] (* Vincenzo Librandi, Oct 28 2017 *)

A294310 Integers with precisely nine partitions into sums of four squares of nonnegative numbers.

Original entry on oeis.org

90, 146, 166, 174, 185, 187, 205, 206, 207, 209, 219, 220, 221, 223, 231, 235, 251, 260, 271, 287, 316, 359, 360, 380, 584, 664, 696, 824, 880, 1040, 1264, 1440, 1520, 2336, 2656, 2784, 3296, 3520, 4160, 5056, 5760, 6080, 9344, 10624, 11136, 13184, 14080

Offset: 1

Robert Price, Oct 27 2017

nonn

A002635(a(n)) = 9.

Robert Price, Table of n, a(n) for n = 1..60
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.
Index entries for sequences related to sums of squares

Cf. A001156, A002635, A006431, A025428, A180149, A245022, A294282, A294297.

f[n_]:=Length@PowersRepresentations[n, 4, 2]; Select[Range@850, f@#==9 &] (* Vincenzo Librandi, Oct 28 2017 *)

A307510 a(n) is the greatest product ijk*l where i^2 + j^2 + k^2 + l^2 = n and 0 <= i <= j <= k <= l.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 4, 0, 3, 8, 0, 6, 16, 0, 12, 4, 9, 24, 8, 18, 0, 16, 36, 12, 32, 0, 24, 54, 0, 48, 20, 36, 81, 40, 72, 30, 64, 0, 60, 108, 45, 96, 40, 90, 48, 80, 144, 60, 135, 72, 120, 54, 0, 192, 108, 180, 96, 160, 72, 162, 256, 144, 240, 100

Offset: 0

Rémy Sigrist, Apr 11 2019

The sequence is well defined as every nonnegative integer can be represented as a sum of four squares in at least one way.

			For n = 34:
- 34 can be expressed in 4 ways as a sum of four squares:
    i^2 + j^2 + k^2 + l^2   i*j*k*l
    ---------------------   -------
    0^2 + 0^2 + 3^2 + 5^2         0
    0^2 + 3^2 + 3^2 + 4^2         0
    1^2 + 1^2 + 4^2 + 4^2        16
    1^2 + 2^2 + 2^2 + 5^2        20
- a(34) = max(0, 16, 20) = 20.

Robert Israel, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Colored scatterplot of the first 20000 terms (where the color is function of the parity of n)
Rémy Sigrist, C program for A307510
Wikipedia, Lagrange's four-square theorem

See A307531 for the additive variant.

Cf. A000534, A002635, A122922.

C
```
See Links section.
```

g:= proc(n, k) option remember; local a;
  if k = 1 then if issqr(n) then return sqrt(n) else return -infinity fi fi;
  max(0,seq(a*procname(n-a^2, k-1), a=1..floor(sqrt(n))))
end proc:
seq(g(n, 4), n=0..100); # Robert Israel, Apr 15 2019

Array[Max[Times @@ # & /@ PowersRepresentations[#, 4, 2]] &, 68, 0] (* Michael De Vlieger, Apr 13 2019 *)

a(n) = 0 iff n belongs to A000534.

a(n) <= (n/4)^2, with equality if and only if n is an even square. - Robert Israel, Apr 15 2019

A002637 Number of partitions of n into not more than 5 pentagonal numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 1, 2, 1, 2, 2, 3, 3, 2, 3, 2, 2, 2, 1, 2, 1, 3, 3, 3, 4, 3, 3, 2, 3, 3, 1, 2, 3, 4, 4, 3, 4, 3, 4, 3, 3, 3, 3, 3, 4, 5, 5, 3, 3, 4, 4, 3, 2, 4, 3, 4, 4, 5, 6, 5, 5, 4, 5, 6, 3, 4, 4, 6, 5, 4, 5, 4, 6, 4, 5, 6, 4, 3, 3, 8, 7, 5, 6, 5, 7, 5, 6, 5, 3, 6, 5, 7, 7

Offset: 1

N. J. A. Sloane

Gino Loria, Sulla scomposizione di un intero nella somma di numeri poligonali. (Italian) Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 1, (1946). 7-15.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Wouter Meeussen, Table of n, a(n) for n = 1..512
D. H. Lehmer, Recent Mathematical Tables (about Loria article), Math. Comp. 2 (1947), 301-302.
Gino Loria, Sulla scomposizione di un intero nella somma di numeri poligonali, (Italian), Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 1, (1946). 7-15. Also D. H. Lehmer, Review of Loria article, Math. Comp. 2 (1947), 301-302. [Annotated scanned copies]
Eric Weisstein's World of Mathematics, Pentagonal Number.

it=Expand[Normal @ Series[CoefficientList[Series[Product[(1+(q l[3k^2/2-k/2] x^(3k^2/2-k/2)))^5,{k,512}],{x,0,512}],x],{q,0,5}]]/. (Integer) q^(e:1)->1 /.q->1 ; Drop[it/.l[]->1,1] (* _Wouter Meeussen, May 17 2008 *)

More terms from Naohiro Nomoto, Feb 28 2002

A294309 Integers with precisely eight partitions into sums of four squares of nonnegative numbers.

Original entry on oeis.org

114, 117, 122, 126, 145, 147, 148, 157, 165, 169, 172, 175, 177, 181, 190, 193, 203, 227, 233, 311, 456, 488, 504, 592, 688, 760, 1824, 1952, 2016, 2368, 2752, 3040, 7296, 7808, 8064, 9472, 11008, 12160, 29184, 31232, 32256, 37888, 44032, 48640

Offset: 1

Robert Price, Oct 27 2017

nonn

A002635(a(n)) = 8.

D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.
Index entries for sequences related to sums of squares

Cf. A001156, A002635, A006431, A025428, A180149, A245022, A294282, A294297.

f[n_]:=Length@PowersRepresentations[n, 4, 2]; Select[Range@850, f@#==8 &] (* Vincenzo Librandi, Oct 28 2017 *)

A294311 Integers with precisely ten partitions into sums of four squares of nonnegative numbers.

Original entry on oeis.org

130, 138, 153, 154, 171, 180, 182, 195, 196, 201, 213, 214, 217, 228, 229, 238, 241, 244, 247, 249, 253, 254, 257, 259, 269, 276, 277, 281, 295, 299, 303, 308, 317, 319, 332, 335, 347, 428, 431, 520, 552, 616, 720, 728, 784, 856, 912, 952, 976, 1016, 1104

Offset: 1

Robert Price, Oct 27 2017

nonn

A002635(a(n)) = 10.

Robert Price, Table of n, a(n) for n = 1..98
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.
Index entries for sequences related to sums of squares

Cf. A001156, A002635, A006431, A025428, A180149, A245022, A294282, A294297, A293175, A294308, A294309, A294310.

f[n_]:=Length@PowersRepresentations[n, 4, 2]; Select[Range@850, f@#==10 &] (* Vincenzo Librandi, Oct 28 2017 *)

cp's OEIS Frontend

A297788 Number of partitions of n into 3 squares and a nonnegative cube.

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A307531 a(n) is the greatest sum i + j + k + l where i^2 + j^2 + k^2 + l^2 = n and 0 <= i <= j <= k <= l.

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A161148 Number of partitions of n such that each term of the partition is a squared divisor of n.

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A293175 Integers with precisely six partitions into sums of four squares of nonnegative numbers.

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A294308 Integers with precisely seven partitions into sums of four squares of nonnegative numbers.

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A294310 Integers with precisely nine partitions into sums of four squares of nonnegative numbers.

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A307510 a(n) is the greatest product i*j*k*l where i^2 + j^2 + k^2 + l^2 = n and 0 <= i <= j <= k <= l.

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C

Maple

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A002637 Number of partitions of n into not more than 5 pentagonal numbers.

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A307510 a(n) is the greatest product ijk*l where i^2 + j^2 + k^2 + l^2 = n and 0 <= i <= j <= k <= l.