A116978 -id:A116978 - cp's OEIS frontend

A118719 Cubes for which the digital root is also a cube.

0, 1, 8, 64, 125, 343, 512, 1000, 1331, 2197, 2744, 4096, 4913, 6859, 8000, 10648, 12167, 15625, 17576, 21952, 24389, 29791, 32768, 39304, 42875, 50653, 54872, 64000, 68921, 79507, 85184, 97336, 103823, 117649, 125000, 140608, 148877

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), May 21 2006

All cubes have a digital root 1,8 or 9. (except for the number 0) So this sequence contains all cubes with a digital root which is not 9.

This sequence is 0 union A016779 union A016791.

			64 is in the sequence because (1) it is a cube and (2) the digital root 1 is also a cube.

Vincenzo Librandi, Table of n, a(n) for n = 1..3000

Cf. A000578, A002117, A010888, A116978.

[0] cat [(6*n+(-1)^n-9)^3 div 64: n in [2..37]];  // Bruno Berselli, May 05 2011

Join[{0}, Table[(3*k + {1, 2})^3, {k, 0, 15}] // Flatten] (* Amiram Eldar, Dec 19 2020 *)

a010888(n)=if(n, (n-1)%9+1)
lista(nn) = {for (n=0, nn, if (ispower(a010888(n^3), 3), print1(n^3, ", ")););} \\ Michel Marcus, Feb 18 2015

a(n) = (floor(3*n/2)-2)^3 for n >= 2. - Nathaniel Johnston, May 05 2011
G.f.: x^2*(1+7*x+53*x^2+40*x^3+53*x^4+7*x^5+x^6)/((1+x)^3*(1-x)^4).  a(n) = A001651(n-1)^3 for n>1. - Bruno Berselli, May 05 2011
Sum_{n>=2} 1/a(n) = 26*zeta(3)/27. - Amiram Eldar, Dec 19 2020

A117678 Squares for which the multiplicative digital root is also a square.

Original entry on oeis.org

0, 1, 4, 9, 25, 100, 169, 196, 225, 256, 400, 529, 576, 625, 676, 900, 961, 1024, 1089, 1156, 1225, 1296, 1521, 1600, 2025, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 2916, 3025, 3136, 3481, 3600, 3844, 3969, 4096, 4225, 4356, 4489, 4900, 5041, 5184, 5329

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), Apr 12 2006

From Robert Israel, Oct 22 2015: (Start)
1, 9, and squares in A034048 and A034051.
Are there infinitely many squares in A034051? (End)

Nathaniel Johnston, Table of n, a(n) for n = 1..5000

Cf. A000290, A031347, A034048, A034051, A116978.

A007954 := proc(n) return mul(d, d=convert(n, base, 10)): end: A117678 := proc(n) option remember: local k, m: if(n=1)then return 0:fi: for k from procname(n-1)+1 do m:=k^2: while(length(m)>1)do m:=A007954(m): od: if(m in {0,1,4,9})then return k: fi: od: end: seq(A117678(n)^2, n=1..47); # Nathaniel Johnston, May 05 2011

Select[Range[0, 73]^2, IntegerQ@ Sqrt[FixedPoint[Times @@ IntegerDigits@ # &, #] &@ #] &] (* Michael De Vlieger, Oct 22 2015 *)

t(k) = {while(k>9, k=prod(i=1, #k=digits(k), k[i])); k}
for(n=0, 100, if(issquare(t(n^2)), print1(n^2, ", "))); \\ Altug Alkan, Oct 22 2015

Offset and some terms corrected by Nathaniel Johnston, May 05 2011

cp's OEIS Frontend

A118719 Cubes for which the digital root is also a cube.

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