login
Repdigit numbers that are divisible by 7.
3

%I #103 Jan 16 2024 07:00:28

%S 0,7,77,777,7777,77777,111111,222222,333333,444444,555555,666666,

%T 777777,888888,999999,7777777,77777777,777777777,7777777777,

%U 77777777777,111111111111,222222222222,333333333333,444444444444,555555555555,666666666666,777777777777

%N Repdigit numbers that are divisible by 7.

%C 7 divides a repdigit iff it consists of only digit 7, or has length 6*k (for any digit).

%C Repdigit remainders A010785(k) mod 7 have period 54. - _Karl-Heinz Hofmann_, Dec 04 2023

%H Karl-Heinz Hofmann, <a href="/A366596/b366596.txt">Table of n, a(n) for n = 1..2329</a>

%H <a href="/index/Rec#order_28">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,1000001,0,0,0,0,0,0,0,0,0,0,0,0,0,-1000000).

%F From _Karl-Heinz Hofmann_, Dec 04 2023: (Start)

%F a(n) = A010785(floor((n-2)/14)*54 + ((n-2) mod 14) + 41), for (n-2) mod 14 > 4.

%F a(n) = (10^(6*floor((n-2)/14) + 6)-1)/9*(((n-2) mod 14)-4), for (n-2) mod 14 > 4.

%F a(n) = A010785(floor((n-2)/14)*54 + ((n-2) mod 14)*9 + 7), for (n-2) mod 14 <= 4.

%F a(n) = (10^(6*floor((n-2)/14) + 1 + ((n-2) mod 14))-1)/9*7, for (n-2) mod 14 <= 4.

%F (End)

%o (PARI) r(n) = 10^((n+8)\9)\9*((n-1)%9+1); \\ A010785

%o lista(nn) = select(x->!(x%7), vector(nn, k, r(k-1))); \\ _Michel Marcus_, Oct 26 2023

%o (Python)

%o def A366596(n):

%o digitlen, digit = (n+12)//14*6, (n+12)%14-4

%o if digit < 1: digitlen += digit - 1; digit = 7

%o return 10**digitlen // 9 * digit # _Karl-Heinz Hofmann_, Dec 04 2023

%Y Intersection of A008589 and A010785.

%Y Cf. A002281 (a subsequence).

%Y Cf. A305322 (divisor 3), A002279 (divisor 5), A083118 (the impossible divisors).

%K nonn,base,easy

%O 1,2

%A _Kritsada Moomuang_, Oct 14 2023