cp's OEIS Frontend

A381699 a(n) is the least nontrivial multiple of 2*n with the least possible number of even digits.

%I A381699 #36 Mar 13 2025 17:19:37
%S A381699 4,8,12,16,30,36,56,32,36,40,110,72,52,56,90,96,136,72,76,80,336,132,
%T A381699 92,96,150,156,378,112,116,120,310,192,132,136,350,576,370,152,156,
%U A381699 160,574,336,172,176,990,552,376,192,196,300,510,312,318,756,330,336,570,1392,354,360,732,372,378
%N A381699 a(n) is the least nontrivial multiple of 2*n with the least possible number of even digits.
%C A381699 By nontrivial multiple, we mean a multiple strictly larger than the number.
%C A381699 For even numbers, the last digit of any multiple will always be even. Also, for multiples of 10^k, all multiples will always have at least k even digits, namely k trailing '0's. Thus, if the number is a multiple of 2*10^k, there will be at least k+1 trailing even digits.
%C A381699 Question: if n is even, but not a multiple of 10, is there always a multiple k*n for which the last digit is the only even digit? If not, what is the smallest counterexample?
%C A381699 Records values of k(n) = a(n)/2n are k(1) = 2, k(5) = 3, k(7) = 4, k(11) = 5, k(21) = 8, k(45) = 11, k(58) = 12, k(101) = 55, k(182) = 108, k(1001) = 555, k(2001) = 778, k(3996) = 1001, k(7992) = 3253, k(9091) = 21545, k(9901) = 161155, ...
%e A381699 a(4) = 16 is the least nontrivial multiple of 8 with only one even digit.
%e A381699 a(5) = 30 is the least nontrivial multiple of 10 with only one even digit.
%e A381699 a(10) = 40 because 40 is the least nontrivial multiple of 20, and all multiples of 20 will always have (at least) the last two digits even.
%e A381699 a(41) = 574 is the least positive multiple of 82 that has only one even digit.
%o A381699 (PARI) apply( {A381699(n, o=valuation(5*n*=2,10))=for(k=2, oo, #[0|d<-digits(n*k)%2, !d]>o|| return(k*n))}, [1..99]) \\ _M. F. Hasler_, Mar 04 2025
%Y A381699 Cf. A061808 (similar for odd numbers), A061807.
%Y A381699 Cf. A118070 (numbers with exactly one even decimal digit).
%K A381699 nonn,base
%O A381699 1,1
%A A381699 _Ali Sada_ and _M. F. Hasler_, Mar 04 2025