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

1,1

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).

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

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 *)

Prog

(Sage) [10*n+2*(-1)^n-5 for n in (1..70)]
(Maxima) makelist(10*n+2*(-1)^n-5, n, 1, 70);
(GAP) List([1..70], n -> 10*n+2*(-1)^n-5);
(Magma) [10*n+2*(-1)^n-5: n in [1..70]];
(Python) [10*n+2*(-1)**n-5 for n in range(1, 70)]
(Julia) [10*n+2*(-1)^n-5 for n in 1:70] |> println
(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

Crossrefs

Cf. A004526, A056849.

Subsequence of A063226, A295009.

Similar sequences are listed in A322489.

Sequence in context: A082372 A273407 A267067 * A351688 A206626 A128107

Adjacent sequences: A322487 A322488 A322489 * A322491 A322492 A322493

Keyword

nonn,base,easy

Author

Bruno Berselli, Dec 12 2018

Status

approved