A322489 Numbers k such that k^k ends with 4.

2, 18, 22, 38, 42, 58, 62, 78, 82, 98, 102, 118, 122, 138, 142, 158, 162, 178, 182, 198, 202, 218, 222, 238, 242, 258, 262, 278, 282, 298, 302, 318, 322, 338, 342, 358, 362, 378, 382, 398, 402, 418, 422, 438, 442, 458, 462, 478, 482, 498, 502, 518, 522, 538, 542, 558

Offset: 1

Author

Bruno Berselli, Dec 12 2018

Keywords

• nonn
• base
• easy

Comments

Also numbers k == 2 (mod 4) such that 2^k and k^2 end with the same digit.

Numbers congruent to {2, 18} 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 A139544, A235700.

Numbers k such that k^k ends with d: A008592 (d=0), A017281 (d=1), A067870 (d=3), this sequence (d=4), A017329 (d=5), A271346 (d=6), A322490 (d=7), A017377 (d=9).

Programs

GAP

List([1..70], n -> 10*n+3*(-1)^n-5);

Julia

[10*n+3*(-1)^n-5 for n in 1:70] |> println

Magma

[10*n+3*(-1)^n-5: n in [1..70]];

Maple

select(n->n^n mod 10=4,[$1..558]); # Paolo P. Lava, Dec 18 2018

Mathematica

Table[10 n + 3 (-1)^n - 5, {n, 1, 60}]

Maxima

makelist(10*n+3*(-1)^n-5, n, 1, 70);

PARI

apply(A322489(n)=10*n+3*(-1)^n-5, [1..70]) \\ M. F. Hasler, Dec 14 2018

PARI

Vec(2*x*(1 + 8*x + x^2) / ((1 - x)^2*(1 + x)) + O(x^70)) \\ Colin Barker, Dec 13 2018

Python

[10*n+3*(-1)**n-5 for n in range(1, 70)]

Sage

[10*n+3*(-1)^n-5 for n in (1..70)]

Formula

O.g.f.: 2*x*(1 + 8*x + x^2)/((1 + x)*(1 - x)^2).

E.g.f.: 2 + 3*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 + 3*(-1)^n - 5. Therefore:

a(n) = 10*n - 8 for odd n;

a(n) = 10*n - 2 for even n.

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

Sum_{n>=1} (-1)^(n+1)/a(n) = tan(2*Pi/5)*Pi/20 = sqrt(5+2*sqrt(5))*Pi/20. - Amiram Eldar, Feb 27 2023