A322490 Numbers k such that k^k ends with 7.

3, 17, 23, 37, 43, 57, 63, 77, 83, 97, 103, 117, 123, 137, 143, 157, 163, 177, 183, 197, 203, 217, 223, 237, 243, 257, 263, 277, 283, 297, 303, 317, 323, 337, 343, 357, 363, 377, 383, 397, 403, 417, 423, 437, 443, 457, 463, 477, 483, 497, 503, 517, 523, 537, 543, 557, 563

Offset: 1

Author

Bruno Berselli, Dec 12 2018

Keywords

• nonn
• base
• easy

Comments

Equivalently, numbers k such that k and (7^h)^k end with the same digit, where h == 1 (mod 4).

Also, numbers k such that k and (3^h)^k end with the same digit, where h == 3 (mod 4).

Numbers congruent to {3, 17} mod 20. - Amiram Eldar, Feb 27 2023

Links

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

Crossrefs

Cf. A004526, A056849.

Subsequence of A063226, A295009.

Similar sequences are listed in A322489.

Programs

GAP

List([1..70], n -> 10*n+2*(-1)^n-5);

Julia

[10*n+2*(-1)^n-5 for n in 1:70] |> println

Magma

[10*n+2*(-1)^n-5: n in [1..70]];

Maple

select(n->n^n mod 10=7,[$1..563]); # Paolo P. Lava, Dec 18 2018

Mathematica

Table[10 n + 2 (-1)^n - 5, {n, 1, 60}]
LinearRecurrence[{1,1,-1},{3,17,23},80] (* Harvey P. Dale, Sep 15 2019 *)

Maxima

makelist(10*n+2*(-1)^n-5, n, 1, 70);

PARI

apply(A322490(n)=10*n+2*(-1)^n-5, [1..70])

PARI

Vec(x*(3 + 14*x + 3*x^2) / ((1 + x)*(1 - x)^2) + O(x^55)) \\ Colin Barker, Dec 13 2018

Python

[10*n+2*(-1)**n-5 for n in range(1, 70)]

Sage

[10*n+2*(-1)^n-5 for n in (1..70)]

Formula

O.g.f.: x*(3 + 14*x + 3*x^2)/((1 + x)*(1 - x)^2).

E.g.f.: 3 + 2*exp(-x) + 5*(2*x - 1)*exp(x).

a(n) = -a(-n+1) = a(n-1) + a(n-2) - a(n-3).

a(n) = 10*n + 2*(-1)^n - 5. Therefore:

a(n) = 10*n - 7 for odd n;

a(n) = 10*n - 3 for even n.

a(n+2*k) = a(n) + 20*k.

Sum_{n>=1} (-1)^(n+1)/a(n) = tan(7*Pi/20)*Pi/20. - Amiram Eldar, Feb 27 2023