A320662 Numbers k for which there are numbers 0 < m <= k such that k^3 + m^3 is a square.

2, 8, 18, 21, 26, 32, 37, 46, 50, 65, 70, 72, 84, 88, 91, 98, 104, 105, 112, 128, 148, 162, 184, 189, 190, 200, 234, 242, 249, 260, 273, 280, 288, 312, 330, 333, 336, 338, 345, 352, 354, 364, 371, 392, 407, 414, 416, 420

Offset: 1

Author

Marius A. Burtea, Oct 18 2018

Keywords

• nonn

Comments

The sequence is infinite since if u is in the sequence then so is u*t^2, t, u >= 1. - Marius A. Burtea and David A. Corneth, Oct 23 2018

For the subsequence k= 8, 18, 32, 50,65, 72, 98, 104, 105,... two or more m exist satisfying the equation. - R. J. Mathar, Jan 22 2025

Examples

			8^3 + 4^3 = 512 + 64 = 576 = 24^2, so 8 is part of the sequence.
18^3 + 9^3 = 5832 + 729 = 6561 = 81^2, so 18 is part of the sequence.
91^3 + 65^3 = 753571 + 274625 = 1028196 = 1014^2, so 91 is part of the sequence.
7^3 + 0^3 = 343 + 0 = 343, 7^3 + 1^3 = 343 + 1 = 344, 7^3 + 2^3 = 343 + 8 = 351,7^3 + 4^3 = 343 + 64 = 407, 7^3 + 5^3 = 343 + 125 = 468, 7^3 + 6^3 = 343 + 216 = 559 and 7^3 + 7^3 = 343 + 343 = 686. Numbers 343, 344, 351, 407, 468, 559 and 686 are not squares, so 7 is not part of the sequence.
		

Crossrefs

Cf. A003325, A003997, A004999, A024670, A086119, A282639 (subsequence for coprime m,k), A050801 (bases of the squares).

Programs

Maple

A320662 := proc(n)
    option remember ;
    local m,k ;
    if n =1 then
        2
    else
        for k from procname(n-1)+1 do
            for m from 1 to k do
                if issqr(k^3+m^3) then
                    return k ;
                end if;
            end do:
        end do:
    end if;
end proc:
seq(A320662(n),n=1..40) ; # R. J. Mathar, Jan 22 2025

Mathematica

Select[Range@ 420, AnyTrue[Range[#1]^3 + #2, IntegerQ@ Sqrt@ # &] & @@ {#, #^3} &] (* Michael De Vlieger, Nov 05 2018 *)

PARI

is(n) = for(m=1, n, if(issquare(n^3+m^3), return(1))); 0 \\ Felix Fröhlich, Oct 22 2018