A133388 Largest integer m such that n-m^2 is a square, or 0 if no such m exists.

1, 1, 0, 2, 2, 0, 0, 2, 3, 3, 0, 0, 3, 0, 0, 4, 4, 3, 0, 4, 0, 0, 0, 0, 5, 5, 0, 0, 5, 0, 0, 4, 0, 5, 0, 6, 6, 0, 0, 6, 5, 0, 0, 0, 6, 0, 0, 0, 7, 7, 0, 6, 7, 0, 0, 0, 0, 7, 0, 0, 6, 0, 0, 8, 8, 0, 0, 8, 0, 0, 0, 6, 8, 7, 0, 0, 0, 0, 0, 8, 9, 9, 0, 0, 9, 0, 0, 0, 8, 9, 0, 0, 0, 0, 0, 0, 9, 7, 0, 10

Offset: 1

Author

M. F. Hasler, Nov 23 2007

Keywords

• nonn

Comments

The sequence could be extended to a(0) = 0.

We could have defined a(n) = -1 instead of 0 if n is not sum of two squares, and then include unambiguously a(0) = 0. At present, a(n) = 0 <=> A000161(n) = 0.

Examples

			a(3) = 0 since 3 cannot be written as sum of 2 perfect squares;
a(5) = 2 since 5 = 2^2 + 1^2.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000161, A001481.

Programs

Maple

a:= proc(n) local t, d;
      for t from 0 do d:= n-t^2;
        if d<0 then break elif issqr(d) then return isqrt(d) fi
      od; 0
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, May 14 2015

Mathematica

a[n_] := Module[{m, d, s}, For[m = 0, True, m++, d = n - m^2; If[d < 0, Break[], s = Sqrt[d]; If[IntegerQ[s], Return[s]]]]; 0];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, May 18 2018, after Alois P. Heinz *)

PARI

sum2sqr(n)={ if(n>1, my(L=List(), f, p=1); for(i=1, matsize(f=factor(n))[1], if(f[i,1]%4==1, listput(L, [qfbsolve(Qfb(1,0,1), f[i,1])*[1, I]~, f[i,2]] ),/*elseif*/ f[i,1]==2, p = (1+I)^f[i,2],/*elseif*/ bittest(f[i,2],0), return([]),/*else*/ p *= f[i,1]^(f[i,2]\2))); L=apply(s->vector(s[2]+1, j, s[1]^(s[2]+1-j)*conj(s[1])^(j-1)), L); my(S=List()); forvec(T=vector(#L, i, [1,#L[i]]), listput(S, prod( j=1, #T, L[j][T[j]] ))); Set(apply(f->vecsort(abs([real(f), imag(f)])), Set(S)*p)), if(n<0, [], [[0, n]]))} \\ updated by M. F. Hasler, May 12 2018. (If PARI version 2.12.x returns an error, append [1] to qfbsolve(...) above. - M. F. Hasler, Dec 12 2019)
apply( A133388=n->if(n=sum2sqr(n),vecmax(Mat(n~))), [1..50]) \\ This sequence: maximum

Python

from sympy.solvers.diophantine.diophantine import diop_DN
def A133388(n): return max((a for a, b in diop_DN(-1,n)),default=0) # Chai Wah Wu, Sep 08 2022

Formula

a(n) = max( sup { max(a,b) | a^2+b^2 = n ; a,b in Z }, 0 )

a(A022544(j))=0, j>0. - R. J. Mathar, Jun 17 2009

a(n^2) = a(n^2 + 1) = n, for all n. Conversely, whenever a(n) = a(n+1), then n = k^2. - M. F. Hasler, Sep 02 2018