A295159 -id:A295159 - cp's OEIS frontend

A295494 Smallest number with exactly n representations as a sum of six nonnegative squares.

0, 4, 9, 17, 20, 30, 29, 38, 36, 45, 52, 53, 54, 65, 74, 68, 83, 77, 90, 84, 86, 99, 100, 107, 101, 108, 110, 117, 129, 116, 131, 125, 126, 146, 152, 140, 134, 192, 156, 149, 161

Offset: 0

Robert Price, Nov 22 2017

nonn

It appears that a(n) does not exist for n in {42, 55, 61, 74, 99, 100, 103, 125, 135, 139, 148, 152, 161, 162, 164, 168, 180, 182, 194, 196}; i.e., there is no integer whose number of representations is any of these values.

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.

D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.

Cf. A000177, A295159.

Table[SelectFirst[Range@ 200, Length@ PowersRepresentations[#, 6, 2] == n &] - Boole[n == 1], {n, 41}] (* Michael De Vlieger, Nov 26 2017 *)

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

Original entry on oeis.org

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

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A295494 Smallest number with exactly n representations as a sum of six nonnegative squares.

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