A321881 Numbers whose sum and product of digits are cubes.

0, 1, 8, 10, 80, 100, 107, 170, 206, 260, 305, 350, 404, 440, 503, 530, 602, 620, 701, 710, 800, 999, 1000, 1007, 1016, 1025, 1034, 1043, 1052, 1061, 1070, 1106, 1124, 1142, 1160, 1205, 1214, 1241, 1250, 1304, 1340, 1403, 1412, 1421, 1430, 1502, 1520, 1601, 1610, 1700

Offset: 1

Author

Enrique Navarrete, Nov 20 2018

Keywords

• nonn
• base

Comments

The first numbers in the sequence that are cubes themselves are 0,1,8,1000,8000.

a(22)=999 is the only term up to n=120 related to the cube 27 (the previous ones relate to 0,1,8).

Also, a(22)=999 is the first term that has more than one digit and consists of a single repeated digit; the next ones are 11111111 and 333333333.

Examples

			93111111111111111 (15 ones) is in the sequence since the sum and the product of the digits is 27 (a cube).
333 is not in the sequence since the product of the digits is 27 but the sum is 9 (not a cube).
		

Links

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

Crossrefs

Cf. A007953, A046031, A062398, A070276, A059094, A237767.

Programs

Magma

[n:n in [0..2000]| IsPower((&+Intseq(n)), 3) and IsPower((&*Intseq(n)), 3)] // Marius A. Burtea, Jan 21 2019

Maple

filter:= proc(n) local L;
  L:= convert(n,base,10);
  simplify(convert(L,`+`)^(1/3))::integer and
  simplify(convert(L,`*`)^(1/3))::integer;
end proc:
select(filter, [$0..1000]); # Robert Israel, Jan 21 2019

Mathematica

cubeQ[n_] := IntegerQ[Surd[n, 3]]; aQ[n_] := cubeQ[Plus @@ IntegerDigits[n]] &&
cubeQ[Times @@ IntegerDigits[n]]; Select[Range[0, 3000], aQ] (* Amiram Eldar, Nov 20 2018 *)

PARI

isok(n) = my(d=digits(n)); ispower(vecsum(d), 3) && ispower(vecprod(d), 3); \\ Michel Marcus, Nov 29 2018