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

1,1

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).

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

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}]

Prog

(Sage) [10*n+3*(-1)^n-5 for n in (1..70)]
(Maxima) makelist(10*n+3*(-1)^n-5, n, 1, 70);
(GAP) List([1..70], n -> 10*n+3*(-1)^n-5);
(Magma) [10*n+3*(-1)^n-5: n in [1..70]];
(Python) [10*n+3*(-1)**n-5 for n in range(1, 70)]
(Julia) [10*n+3*(-1)^n-5 for n in 1:70] |> println
(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

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).

Sequence in context: A092587 A247457 A015787 * A063430 A031104 A115042

Adjacent sequences: A322486 A322487 A322488 * A322490 A322491 A322492

Keyword

nonn,base,easy

Author

Bruno Berselli, Dec 12 2018

Status

approved