A295159 Smallest number with exactly n representations as a sum of five nonnegative squares.

0, 4, 13, 20, 29, 37, 50, 52, 61, 74, 77, 85, 91, 101, 106, 118, 125, 131, 133, 139, 162, 157, 154, 166, 178, 194, 187, 205, 203, 202, 227, 211, 226, 235, 234, 269, 251, 275, 250, 266, 291, 274, 259, 283, 301, 325, 306, 298, 326, 334, 347, 322, 362, 447, 331

Offset: 1

Author

Robert Price, Nov 15 2017

Keywords

• nonn

Comments

Conjecture: a(448) does not exist, i.e., there is no number with exactly 448 such representations. - Robert Israel, Nov 15 2017

References

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

Links

• Robert Israel, Table of n, a(n) for n = 1..447 (first 200 terms from Robert Price)
• D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.

Crossrefs

Cf. A000174, A006431, A294675.

Programs

Maple

N:= 1000: # to get a(1)...a(n) where a(n+1) is the first term > N
V:= Array(0..N):
for x[1] from 0 to floor(sqrt(N/5)) do
  for x[2] from x[1] while x[1]^2 + 4*x[2]^2 <= N do
    for x[3] from x[2] while x[1]^2 + x[2]^2 + 3*x[3]^2 <= N do
      for x[4] from x[3] while x[1]^2 + x[2]^2 + x[3]^2 + 2*x[4]^2 <= N do
        for x[5] from x[4] while x[1]^2 + x[2]^2 + x[3]^2 + x[4]^2 + x[5]^2 <= N do
           t:=  x[1]^2 + x[2]^2 + x[3]^2 + x[4]^2 + x[5]^2;
           V[t]:= V[t]+1;
od od od od od:
A:= Vector(max(V),-1):
for i from 0 to N do if A[V[i]]=-1 then A[V[i]]:= i fi od:
T:= select(t -> A[t]=-1, [$1..max(V)]):
if T = [] then nmax:= max(V) else nmax:= T[1]-1 fi:
convert(A[1..nmax],list); # Robert Israel, Nov 15 2017

Formula

A000174(a(n))=n. - Robert Israel, Nov 15 2017