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

1,2

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.

Formula

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

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

Crossrefs

Cf. A000174, A006431, A294675.

Sequence in context: A299182 A030744 A173512 * A298216 A299092 A299874

Adjacent sequences: A295156 A295157 A295158 * A295160 A295161 A295162

Keyword

nonn

Author

Robert Price, Nov 15 2017

Status

approved