A002635 -id:A002635 - cp's OEIS frontend

A295218 Number of partitions of 2*n-1 into four squares.

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

Offset: 1

Franklin T. Adams-Watters, Nov 17 2017

nonn

This is a bisection of A002635.
While A002635 contains each positive integer infinitely often, here a number can appear only finitely many times.
By the Jacobi theorem, a(n) >= A000203(n)/48 >= (1+n)/48, which implies the previous comment. - Robert Israel, Nov 21 2017

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Jacobi four square theorem

Cf. A002635, A295159.

N:= 100: # to get a(1)...a(N)
V:= Array(0..2*N-1):
for a from 0 while 4*a^2 <= 2*N-1 do
  for b from a while a^2 + 3*b^2 <= 2*N-1 do
     for c from b while a^2 + b^2 + 2*c^2 <= 2*N-1 do
       for d from c while a^2 + b^2 + c^2 + d^2 <= 2*N-1 do
         t:= a^2 + b^2 + c^2 + d^2;
         V[t]:= V[t]+1
od od od od:
seq(V[2*i-1],i=1..N); # Robert Israel, Nov 21 2017

A297930 Number of partitions of n into 2 squares and 2 nonnegative cubes.

Original entry on oeis.org

1, 2, 3, 2, 2, 2, 2, 1, 2, 4, 5, 3, 2, 3, 2, 1, 3, 5, 6, 3, 3, 3, 2, 0, 2, 5, 6, 5, 4, 5, 2, 2, 4, 5, 6, 4, 6, 6, 4, 2, 4, 6, 4, 4, 4, 7, 3, 2, 4, 3, 5, 4, 7, 8, 5, 3, 3, 3, 5, 5, 5, 6, 4, 3, 6, 7, 8, 7, 5, 7, 4, 2, 7, 9, 10, 4, 5, 7, 3, 3, 9, 10, 8, 5, 4, 7

Offset: 0

XU Pingya, Jan 08 2018

nonn

For n <= 6 * 10^7, except for a(23) = 0, all a(n) > 0.

First occurrence of k beginning with 0: 23, 7, 1, 2, 9, 10, 18, 45, 53, 73, 74, 101, 125, 146, 165, 197, ..., . - Robert G. Wilson v, Jan 14 2018

			2 = 0^2 + 0^2 + 1^3 + 1^3 = 0^2 + 1^2 + 0^3 + 1^3 = 1^2 + 1^2 + 0^3 + 0^3, a(2) = 3.
10 = 0^2 + 1^2 + 1^3 + 2^3 = 0^2 + 3^2 + 0^3 + 1^3 = 1^2 + 1^2 + 0^3 + 2^3 = 1^2 + 3^2 + 0^3 + 0^3 = 2^2 + 2^2 + 1^3 + 1^3, a(10) = 5.

W. Jagy and I. Kaplansky, Sums of Squares, Cubes and Higher Powers, Experimental Mathematics, vol. 4 (1995) pp. 169-173.
Eric Weisstein's World of Mathematics, Waring's Problem

Cf. A000164, A002635, A022552, A274274.

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

A357069 Number of partitions of n into at most 4 distinct positive squares.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Oct 25 2022

nonn

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

Cf. A000290, A002635, A025435, A025443, A033461, A341040, A347534, A347586, A347711.

a(n) = Sum_{k=0..4} A341040(n,k). - Alois P. Heinz, Oct 25 2022

A294081 Number of partitions of n into three squares and two nonnegative 7th powers.

Original entry on oeis.org

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

Offset: 0

XU Pingya, Feb 09 2018

nonn

4^i(8j + 7) - 1^7 - 1^7 == 5 (mod 8) (when i = 0), or 2 (when i = 1), or 6 (when i >= 2). Thus, each nonnegative integer can be written as a sum of three squares and two nonnegative 7th powers; i.e., a(n) > 0.

More generally, each nonnegative integer can be written as a sum of three squares and a nonnegative k-th power and a nonnegative m-th power.

			7 = 0^2 + 1^2 + 2^2 + 1^7 + 1^7 = 1^2 + 1^1 + 2^2 + 0^7 + 1^7, a(7) = 2.
10 = 0^2 + 0^2 + 3^2 + 0^7 + 1^7 = 0^2 + 1^1 + 3^2 + 0^7 + 0^7 = 0^2 + 2^2 + 2^2 + 1^7 + 1^7 = 1^2 + 2^1 + 2^2 + 0^7 + 1^7, a(10) = 4.

Wikipedia, Legendre's three-square theorem
Wikipedia, Lagrange's four-square theorem

Cf. A000164, A002635, A004215, A004771, A297970.

a[n_]:=Sum[If[x^2+y^2+z^2+u^7+v^7==n, 1, 0], {x,0,n^(1/2)}, {y,x,(n-x^2)^(1/2)}, {z,y,(n-x^2-y^2)^(1/2)}, {u,0,(n-x^2-y^2-z^2)^(1/7)}, {v,u,(n-x^2-y^2-z^2-u^7)^(1/7)}]
Table[a[n], {n,0,86}]

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A295218 Number of partitions of 2*n-1 into four squares.

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A297930 Number of partitions of n into 2 squares and 2 nonnegative cubes.

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A357069 Number of partitions of n into at most 4 distinct positive squares.

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A294081 Number of partitions of n into three squares and two nonnegative 7th powers.

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