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Smallest multiple of n using a single digit with multiplicity, or 0 if no such number exists.
3

%I #17 Jan 23 2024 17:12:08

%S 1,2,3,4,5,6,7,8,9,0,11,444,111111,222222,555,0,1111111111111111,666,

%T 111111111111111111,0,777,22,1111111111111111111111,888,0,222222,999,

%U 444444,1111111111111111111111111111,0,111111111111111,0,33,2222222222222222,555555

%N Smallest multiple of n using a single digit with multiplicity, or 0 if no such number exists.

%C 1. If p is a prime > 5 then there exists a d such that a(p) = concatenation of '1' d times where p = k*d + 1 for some k. a(p)= (10^d -1)/9 < ={10^(p-1)- 1}/9.

%C 2. a(n) = 0 if n = 10k, 16k or 25k.

%D Amarnath Murthy, "On the divisors of the Smarandache Unary sequence," Smarandache Notions Journal, Volume 11, 1-2-3, Spring 2000.

%H Bo Gyu Jeong, <a href="/A083116/b083116.txt">Table of n, a(n) for n = 1..1000</a>

%H Bo Gyu Jeong, <a href="/A083116/a083116.cpp.txt">C++ code which computes A083117 and this sequence</a>

%o (Python)

%o from itertools import count

%o def A083116(n):

%o if not (n%10 and n%16 and n%25): return 0

%o for l in count(1):

%o k = (10**l-1)//9

%o for a in range(1,10):

%o if not (m:=a*k)%n:

%o return m # _Chai Wah Wu_, Jan 23 2024

%Y Cf. A083117, A083118.

%K base,nonn

%O 1,2

%A _Amarnath Murthy_ and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 23 2003

%E a(21) corrected by _Bo Gyu Jeong_, Jun 12 2012

%E More terms from _Bo Gyu Jeong_, Jun 13 2012