A067870 Numbers k such that k and 3^k end with the same digit.

7, 13, 27, 33, 47, 53, 67, 73, 87, 93, 107, 113, 127, 133, 147, 153, 167, 173, 187, 193, 207, 213, 227, 233, 247, 253, 267, 273, 287, 293, 307, 313, 327, 333, 347, 353, 367, 373, 387, 393, 407, 413, 427, 433, 447, 453, 467, 473, 487, 493, 507, 513, 527, 533

Offset: 1

Author

Benoit Cloitre, Mar 07 2002

Keywords

• nonn
• base
• easy

Comments

Also numbers k such that k^k ends with 3. - Bruno Berselli, Dec 11 2018

Numbers congruent to {7, 13} mod 20. - Amiram Eldar, Feb 27 2023

Examples

			3^13 = 1594323 hence 13 is in the sequence.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Programs

Mathematica

LinearRecurrence[{1, 1, -1}, {7, 13, 27}, 50] (* Amiram Eldar, Feb 27 2023 *)

PARI

a(n) = (5*(2*n-1)*(-1)^n - 2)*(-1)^n; \\ Jinyuan Wang, Apr 06 2020

PARI

Vec(x*(7 + 6*x + 7*x^2) / ((1 - x)^2*(1 + x)) + O(x^50)) \\ Colin Barker, Apr 06 2020

Formula

a(2*n+1) = 20*n-13, a(2*n) = 20*n-7.

a(n) = 20*(n-1)-a(n-1) for n>1, a(1)=7. - Vincenzo Librandi, Aug 08 2010

From Colin Barker, Apr 06 2020: (Start)

G.f.: x*(7 + 6*x + 7*x^2) / ((1 - x)^2*(1 + x)).

a(n) = -5 - 2*(-1)^n + 10*n for n>0.

a(n) = a(n-1) + a(n-2) - a(n-3) for n>3. (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = tan(3*Pi/20)*Pi/20. - Amiram Eldar, Feb 27 2023