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

%I A295159 #14 Apr 29 2021 00:54:56
%S A295159 0,4,13,20,29,37,50,52,61,74,77,85,91,101,106,118,125,131,133,139,162,
%T A295159 157,154,166,178,194,187,205,203,202,227,211,226,235,234,269,251,275,
%U A295159 250,266,291,274,259,283,301,325,306,298,326,334,347,322,362,447,331
%N A295159 Smallest number with exactly n representations as a sum of five nonnegative squares.
%C A295159 Conjecture: a(448) does not exist, i.e., there is no number with exactly 448 such representations. - _Robert Israel_, Nov 15 2017
%D A295159 E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.
%H A295159 Robert Israel, <a href="/A295159/b295159.txt">Table of n, a(n) for n = 1..447</a> (first 200 terms from Robert Price)
%H A295159 D. H. Lehmer, <a href="http://www.jstor.org/stable/2305380">On the Partition of Numbers into Squares</a>, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.
%F A295159 A000174(a(n))=n. - _Robert Israel_, Nov 15 2017
%p A295159 N:= 1000: # to get a(1)...a(n) where a(n+1) is the first term > N
%p A295159 V:= Array(0..N):
%p A295159 for x[1] from 0 to floor(sqrt(N/5)) do
%p A295159   for x[2] from x[1] while x[1]^2 + 4*x[2]^2 <= N do
%p A295159     for x[3] from x[2] while x[1]^2 + x[2]^2 + 3*x[3]^2 <= N do
%p A295159       for x[4] from x[3] while x[1]^2 + x[2]^2 + x[3]^2 + 2*x[4]^2 <= N do
%p A295159         for x[5] from x[4] while x[1]^2 + x[2]^2 + x[3]^2 + x[4]^2 + x[5]^2 <= N do
%p A295159            t:=  x[1]^2 + x[2]^2 + x[3]^2 + x[4]^2 + x[5]^2;
%p A295159            V[t]:= V[t]+1;
%p A295159 od od od od od:
%p A295159 A:= Vector(max(V),-1):
%p A295159 for i from 0 to N do if A[V[i]]=-1 then A[V[i]]:= i fi od:
%p A295159 T:= select(t -> A[t]=-1, [$1..max(V)]):
%p A295159 if T = [] then nmax:= max(V) else nmax:= T[1]-1 fi:
%p A295159 convert(A[1..nmax],list); # _Robert Israel_, Nov 15 2017
%Y A295159 Cf. A000174, A006431, A294675.
%K A295159 nonn
%O A295159 1,2
%A A295159 _Robert Price_, Nov 15 2017