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

1,1

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

Table of n, a(n) for n=1..50.

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

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

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

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

Crossrefs

Cf. A010888, A153130, A010718.

Sequence in context: A042603 A031120 A029450 * A299124 A278415 A063243

Adjacent sequences: A225536 A225537 A225538 * A225540 A225541 A225542

Keyword

nonn,base,easy

Author

Marcus Hedbring, May 17 2013

Status

approved