A215537 Lowest k such that k is representable as both the sum of n and of n+1 nonzero squares.

25, 17, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Author

Jon Perry, Aug 15 2012

Keywords

• nonn

Examples

			25 = 5^2 = 3^2 + 4^2
17 = 4^2 + 1^2 = 3^2 + 2^2 + 2^2
12 = 2^2 + 2^2 + 2^2 = 3^2 + 1^2 + 1^2 + 1^2
after this just add 1^2 to both sides.
		

Crossrefs

Cf. A000290 (representable as sum of 1 square), A000404 (sum of 2 positive squares), A000408 (sum of 3 positive squares), A000414 (sum of 4 positive squares), A047700 (sum of 5 positive squares)

Programs

Maple

# true if a is representable as a sum of n squares, each square >= m^2.
isRepnSqrsMin := proc(a,n,m)
    local mpr ;
    if a < n*m^2 then
        return false;
    end if;
    if n = 1 then
        if a>= m^2 and issqr(a) then
            true;
        else
            false;
        end if;
    else
        for mpr from m to a do
            if a-mpr^2 < 1 then
                return false;
            elif procname(a-mpr^2,n-1,mpr) then
                return true;
            end if;
        end do:
    end if;
end proc:
# true if a is representable as a sum of n positive squares.
isRepnSqrs := proc(a,n)
    isRepnSqrsMin(a,n,1) ;
end proc:
A215537 := proc(n)
    local k;
    for k from 1 do
        if isRepnSqrs(k,n) and isRepnSqrs(k,n+1) then
            return k;
        end if;
    end do:
end proc: # R. J. Mathar, Sep 11 2012