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A116978 Cubes whose multiplicative digital root is also a cube.

%I A116978 #34 Feb 16 2025 08:33:00
%S A116978 0,1,8,64,125,343,512,1000,4096,4913,5832,6859,8000,9261,10648,13824,
%T A116978 15625,17576,19683,21952,27000,32768,35937,39304,42875,46656,50653,
%U A116978 54872,59319,64000,68921,74088,79507,85184,91125,97336,103823,110592,117649
%N A116978 Cubes whose multiplicative digital root is also a cube.
%C A116978 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
%H A116978 Nathaniel Johnston, <a href="/A116978/b116978.txt">Table of n, a(n) for n = 1..5000</a>
%H A116978 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MultiplicativeDigitalRoot.html">Multiplicative Digital Root.</a>
%F A116978 a(n) >= A052044(n)^3 for n > 3. - _Charles R Greathouse IV_, Nov 17 2015
%p A116978 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
%t A116978 fQ[n_] := IntegerQ[ FixedPoint[Times @@ IntegerDigits@# &, n]^(1/3)]; Select[Range[0, 48]^3, fQ@# &] (* _Robert G. Wilson v_, Apr 03 2006 *)
%o A116978 (PARI) t(k)=while(k>9, k=prod(i=1, #k=digits(k), k[i])); k
%o A116978 for(n=0, 200, if(ispower(t(n^3), 3), print1(n^3, ", "))); \\ _Altug Alkan_, Oct 22 2015
%Y A116978 Cf. A000578, A031347, A117678, A052044.
%K A116978 nonn,easy,base
%O A116978 1,3
%A A116978 Luc Stevens (lms022(AT)yahoo.com), Apr 01 2006
%E A116978 Corrected and extended by _Robert G. Wilson v_, Apr 03 2006