A225539 Numbers n where 2^n and n have the same digital root.

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

Offset: 1

Author

Marcus Hedbring, May 17 2013

Keywords

• nonn
• base
• easy

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).

Examples

			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.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A010888, A153130, A010718.

Programs

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 *)

PARI

forstep(n=16,500,[7,11],print1(n", ")) \\ Charles R Greathouse IV, May 19 2013

Formula

a(n) = 9*n - 3 + (-1)^n.

a(n) = a(n-1) + 7 (odd n), a(n) = a(n-1) + 11 (even n) with a(1) = 5.

G.f. x*(5 + 11*x + 2*x^2) / ((1-x)^2 * (1+x)). - Joerg Arndt, May 17 2013