A307417 Numbers that can be expressed in a base in such a way that the sum of cubes of their digits in this base equals the original number.

8, 9, 16, 17, 27, 28, 29, 35, 43, 54, 55, 62, 64, 65, 72, 91, 92, 99, 118, 125, 126, 127, 128, 133, 134, 152, 153, 189, 190, 216, 217, 224, 243, 244, 250, 251, 280, 307, 341, 342, 343, 344, 351, 370, 371, 407, 432, 433, 468, 469, 512, 513, 514, 520, 539, 559

Offset

1,1

Comments

There are infinitely many such numbers (proof in the second Johnson link).

Links

César Eliud Lozada, Table of n, a(n) for n = 1..114

Allan Wm. Johnson Jr., Crux Mathematicorum, Vol. 5, No. 1, January 1979, problem 407, 16.

Allan Wm. Johnson Jr., Crux Mathematicorum, Vol. 5, No. 9, November 1979, solution to problem 407, 273-277.

Example

a(1) = 8 = [2, 0] (base 4) =  2^3 + 0^3
a(2) = 9 = [2, 1] (base 4) =  2^3 + 1^3
a(3) = 16 = [2, 2] (base 7) =  2^3 + 2^3
a(4) = 17 = [1, 2, 2] (base 3) =  1^3 + 2^3 + 2^3

Maple

sqn:= []; lis:=[];
for n to 1000 do
  b := 2;
  while b < n do #needs to be adjusted
    q := convert(n, base, b);
    s := convert(map(proc (X) options operator, arrow; X^3 end proc, q), `+`);
    if evalb(s = n) then
      sqn := [op(sqn), n];
      lis := [op(lis), [n, b, ListTools[Reverse](q)]];
      break
    end if;
    b := b+1
  end do
end do;
lis := lis; #list of decompositions [number, base, conversion]
sqn := sqn; #sequence

Crossrefs

Cf. A005188, A023052, A046197.

Sequence in context: A265732 A047393 A241263 * A037371 A236300 A046790

Adjacent sequences: A307414 A307415 A307416 * A307418 A307419 A307420

Keyword

nonn,base

Author

César Eliud Lozada, Apr 07 2019

Status

approved