A271346 Numbers k such that the final digit of k^k is 6.

4, 6, 8, 12, 14, 16, 24, 26, 28, 32, 34, 36, 44, 46, 48, 52, 54, 56, 64, 66, 68, 72, 74, 76, 84, 86, 88, 92, 94, 96, 104, 106, 108, 112, 114, 116, 124, 126, 128, 132, 134, 136, 144, 146, 148, 152, 154, 156, 164, 166, 168, 172, 174, 176, 184, 186, 188, 192, 194

Offset

1,1

Comments

The values of n^n (A000312) end in every digit except for 2 and 8. The sequence of final digits of n^n (A056849) is periodic with period 20; for n=1,2,... the last digits are [1, 4, 7, 6, 5, 6, 3, 6, 9, 0, 1, 6, 3, 6, 5, 6, 7, 4, 9, 0]. Thus, 6 is the most common final digit of n^n. Since 6 does not occur at any odd index in the list above, all terms of a(n) are even. Also, from the distribution of 6's in the list, we can see that the difference between any two consecutive values of a(n) will be 2, 4 or 8.

Links

Felix Fröhlich, Table of n, a(n) for n = 1..10000

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Formula

a(n) = a(n-1) + a(n-6) - a(n-7) for n > 7. - Wesley Ivan Hurt, Oct 08 2017

G.f.: 2*x*(1 + x^2)*(2 + x - x^2 + x^3 + 2*x^4) / ((1 - x)^2*(1 + x)*(1 - x + x^2)*(1 + x + x^2)). - Colin Barker, Dec 13 2018

Maple

A271346:=n->`if`(n^n mod 10 = 6, n, NULL): seq(A271346(n), n=1..500);

Mathematica

LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {4, 6, 8, 12, 14, 16, 24}, 59] (* Ray Chandler, Mar 08 2017 *)

Prog

(PARI) is(n) = Mod(n, 10)^n==6 \\ Felix Fröhlich, Apr 07 2016
(Magma) I:=[4, 6, 8, 12, 14, 16, 24]; [n le 7 select I[n] else Self(n-1)+Self(n-6)-Self(n-7): n in [1..60]]; // Vincenzo Librandi, Oct 09 2017
(PARI) Vec(2*x*(1 + x^2)*(2 + x - x^2 + x^3 + 2*x^4) / ((1 - x)^2*(1 + x)*(1 - x + x^2)*(1 + x + x^2)) + O(x^59)) \\ Colin Barker, Dec 13 2018

Crossrefs

Cf. A000312 (n^n), A056849 (final digit of n^n).

Sequence in context: A235036 A107303 A028876 * A053579 A074121 A175088

Adjacent sequences: A271343 A271344 A271345 * A271347 A271348 A271349

Keyword

nonn,base,easy

Author

Wesley Ivan Hurt, Apr 04 2016

Status

approved