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

Author

Wesley Ivan Hurt, Apr 04 2016

Keywords

• nonn
• base
• easy

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).

Crossrefs

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

Programs

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

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 *)

PARI

is(n) = Mod(n, 10)^n==6 \\ Felix Fröhlich, Apr 07 2016

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

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