A299173 a(n) is the maximum number of squared consecutive positive integers into which the integer n can be partitioned.

1, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 2, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 4, 5, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 2, 4, 0, 0, 0, 5, 6, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Author

Jean-François Alcover, Feb 04 2018

Keywords

• nonn
• look

Comments

a(k^2)>=1, the inequality being strict if k is in A097812.

Examples

			25 = 5^2 = 3^2 + 4^2 and no such partition is longer, so a(25) = 2.
30 = 1^2 + 2^2 + 3^2 + 4^2 and no such partition is longer, so a(30) = 4.
2018 = 7^2 + 8^2 + 9^2 + 10^2 + 11^2 + 12^2 + 13^2 + 14^2 + 15^2 + 16^2 + 17^2 + 18^2 and no such partition is longer, so a(2018) = 12. (This special example is due to _Seiichi Manyama_.) - _Jean-François Alcover_, Feb 05 2018
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A034705, A097812, A130052, A111044, A234304, A234311, A296338, A298467.

Programs

Maple

N:= 200: # to get a(1)..a(N)
A:= Vector(N):
S:= n -> n*(n+1)*(2*n+1)/6:
M:= floor(sqrt(N)):
for d from 1 to M do
  for b from d to M do
    s:= S(b) - S(b-d);
    if s > N then break fi;
    A[s]:= d
od od:
convert(A,list); # Robert Israel, Feb 04 2018

Mathematica

terms = 100; jmax = Ceiling[Sqrt[terms]]; kmax = Ceiling[(3*terms)^(1/3)]; Clear[a]; a[_] = 0; Do[r = Range[j, j + k - 1]; n = r . r; If[k > a[n], a[n] = k], {j, jmax}, {k, kmax}]; Array[a, terms]