A116978 Cubes whose multiplicative digital root is also a cube.

0, 1, 8, 64, 125, 343, 512, 1000, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 13824, 15625, 17576, 19683, 21952, 27000, 32768, 35937, 39304, 42875, 46656, 50653, 54872, 59319, 64000, 68921, 74088, 79507, 85184, 91125, 97336, 103823, 110592, 117649

Offset: 1

Author

Luc Stevens (lms022(AT)yahoo.com), Apr 01 2006

Keywords

• nonn
• easy
• base

Comments

Presumably a(n) ~ n^3. Up to 10^6 there are 14 missing cubes, up to 10^9 there are 36, up to 10^12 there are 81, up to 10^15 there are 155, up to 10^18 there are 267, up to 10^21 there are 517, and up to 10^24 there are 846. - Charles R Greathouse IV, Nov 17 2015

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..5000
• Eric Weisstein's World of Mathematics, Multiplicative Digital Root.

Crossrefs

Cf. A000578, A031347, A117678, A052044.

Programs

Maple

A007954 := proc(n) return mul(d, d=convert(n, base, 10)): end: A116978 := proc(n) option remember: local k,m: if(n=1)then return 0:fi: for k from procname(n-1)+1 do m:=k^3: while(length(m)>1)do m:=A007954(m): od: if(m in {0,1,8})then return k: fi: od: end: seq(A116978(n)^3, n=1..50); # Nathaniel Johnston, May 05 2011

Mathematica

fQ[n_] := IntegerQ[ FixedPoint[Times @@ IntegerDigits@# &, n]^(1/3)]; Select[Range[0, 48]^3, fQ@# &] (* Robert G. Wilson v, Apr 03 2006 *)

PARI

t(k)=while(k>9, k=prod(i=1, #k=digits(k), k[i])); k
for(n=0, 200, if(ispower(t(n^3), 3), print1(n^3, ", "))); \\ Altug Alkan, Oct 22 2015

Formula

a(n) >= A052044(n)^3 for n > 3. - Charles R Greathouse IV, Nov 17 2015

Extensions

Corrected and extended by Robert G. Wilson v, Apr 03 2006