

A225539


Numbers n where 2^n and n have the same digital root.


0



5, 16, 23, 34, 41, 52, 59, 70, 77, 88, 95, 106, 113, 124, 131, 142, 149, 160, 167, 178, 185, 196, 203, 214, 221, 232, 239, 250, 257, 268, 275, 286, 293, 304, 311, 322, 329, 340, 347, 358, 365, 376, 383, 394, 401, 412, 419, 430, 437, 448
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

The digital roots of n have a cycle length of 9 (A010888) and the digital roots of 2^n have a cycle length of 6 (A153130). Therefore, if n is a term so is n+18.
The only values of the digital roots of a(n) are 5 and 7 (A010718).


LINKS



FORMULA

a(n) = 9*n  3 + (1)^n.
a(n) = a(n1) + 7 (odd n), a(n) = a(n1) + 11 (even n) with a(1) = 5.
G.f. x*(5 + 11*x + 2*x^2) / ((1x)^2 * (1+x)).  Joerg Arndt, May 17 2013


EXAMPLE

For n=23, the digital root of n is 5. 2^n equals 8388608 so the digital root of 2^n is 5 as well.


MATHEMATICA

digitalRoot[n_] := Module[{r = n}, While[r > 9, r = Total[IntegerDigits[ r]]]; r]; Select[Range[448], digitalRoot[2^#] == digitalRoot[#] &] (* T. D. Noe, May 19 2013 *)
LinearRecurrence[{1, 1, 1}, {5, 16, 23}, 60] (* Harvey P. Dale, Dec 29 2018 *)


PROG



CROSSREFS



KEYWORD

nonn,base,easy


AUTHOR



STATUS

approved



