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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

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

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Last modified December 3 16:06 EST 2022. Contains 358534 sequences. (Running on oeis4.)