A332171 a(n) = 7*(10^(2n+1)-1)/9 - 6*10^n.

1, 717, 77177, 7771777, 777717777, 77777177777, 7777771777777, 777777717777777, 77777777177777777, 7777777771777777777, 777777777717777777777, 77777777777177777777777, 7777777777771777777777777, 777777777777717777777777777, 77777777777777177777777777777, 7777777777777771777777777777777

Offset

0,2

Comments

For n == 0 or n == 2 (mod 6), there is no obvious divisibility pattern.

According to M. Kamada, n = 116 is the only index of a prime up to n = 10^5.

Links

Table of n, a(n) for n=0..15.

Makoto Kamada, Factorization of 77...77177...77.

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Formula

a(n) = 7*A138148(n) + 10^n.

For n == 1 (mod 3), 3 | a(n) and a(n)/3 = 259*(10^(2n+1)-1)/999 - 2*10^n;

for n == 3 or 5 (mod 6), 13 | a(n) and a(n)/13 = (A(n)-1)*10^n + B(n), where A(n) (resp. B(n)) are the n leftmost (resp. rightmost) digits of 59829*(10^(ceiling(n/6)*6)-1)/(10^6-1).

From Colin Barker, Feb 07 2020: (Start)

G.f.: (1 + 606*x - 1300*x^2) / ((1 - x)*(1 - 10*x)*(1 - 100*x)).

a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n>2.

(End)

E.g.f.: (1/9)*exp(x)*(70*exp(99*x) - 54*exp(9*x) - 7). - Stefano Spezia, Feb 08 2020

Mathematica

Array[7 (10^(2 # + 1) - 1)/9 - 6*10^# &, 15, 0] (* or *)
CoefficientList[Series[(1 + 606 x - 1300 x^2)/((1 - x) (1 - 10 x) (1 - 100 x)), {x, 0, 15}], x] (* Michael De Vlieger, Feb 08 2020 *)
Table[FromDigits[Join[PadRight[{}, n, 7], {1}, PadRight[{}, n, 7]]], {n, 0, 20}] (* or *) LinearRecurrence[ {111, -1110, 1000}, {1, 717, 77177}, 20] (* Harvey P. Dale, Apr 04 2024 *)

Prog

(PARI) apply( {A332171(n)=10^(n*2+1)\9*7-6*10^n}, [0..15])
(PARI) Vec((1 + 606*x - 1300*x^2) / ((1 - x)*(1 - 10*x)*(1 - 100*x)) + O(x^15)) \\ Colin Barker, Feb 07 2020
(Python) def A332171(n): return 10**(n*2+1)//9*7-6*10^n

Crossrefs

Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).

Cf. A002275 (repunits R_n = [10^n/9]), A002281 (7*R_n), A011557 (10^n).

Cf. A332121 .. A332191 (variants with different repeated digit 2, ..., 9).

Cf. A332170 .. A332179 (variants with different middle digit 2, ..., 9).

Sequence in context: A233629 A151631 A183841 * A220724 A034587 A111416

Adjacent sequences: A332168 A332169 A332170 * A332172 A332173 A332174

Keyword

nonn,base,easy

Author

M. F. Hasler, Feb 06 2020

Status

approved