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A067044 Smallest positive k such that k*n contains only even digits.

%I A067044 #32 Mar 09 2025 12:50:32
%S A067044 2,1,2,1,4,1,4,1,32,2,2,2,2,2,4,3,4,16,12,1,2,1,2,1,8,1,18,1,14,2,2,2,
%T A067044 2,2,8,8,6,6,12,1,2,1,2,1,64,1,6,1,14,4,4,4,8,9,4,4,4,7,14,1,4,1,14,1,
%U A067044 4,1,4,1,12,4,4,4,28,3,8,3,6,6,34,1,6,1,8,1,8,1,24,1,32,32,22,5,22,3
%N A067044 Smallest positive k such that k*n contains only even digits.
%C A067044 No multiple of 10 can appear in this sequence. - _M. F. Hasler_, Mar 07 2025
%H A067044 Paul Tek, <a href="/A067044/b067044.txt">Table of n, a(n) for n = 1..10000</a>
%F A067044 From _M. F. Hasler_, Mar 07 2025: (Start)
%F A067044 There is an explicit formula for many values of n:
%F A067044 a(n) = 1 if n has only even digits <=> n is in A014263, else:
%F A067044 a(n) = 2 if n has only digits < 5 <=> n is in A007091;
%F A067044 a(m*(10^k-1)) = 8*round(10^k/6)^2/m for m = 1, 2, 4 or 8 and any k > 0;
%F A067044 a(5*(10^k-1)) = 16*round(10^k/6)^2 for any k > 0;
%F A067044 a(50*m + {5 or 15}) = 4 if m has all digits < 5. (End)
%e A067044 a(7) = 4 as among the multiples of 7 (i.e., 7, 14, 21, 28...), 28 is the smallest multiple with only even digits and a(7)= 28/7 = 4.
%e A067044 a(16) = 3 is the first odd term > 1, a(n = 54, 58, 74, 76, 92, 94, 96, 98, ...) are the next examples, cf. A380874. - _M. F. Hasler_, Mar 03 2025
%t A067044 Table[k = n; While[Length[Intersection[{1, 3, 5, 7, 9}, IntegerDigits[k]]] > 0, k = k + n]; k/n, {n, 100}] (* _T. D. Noe_, Jun 03 2013 *)
%t A067044 sk[n_]:=Module[{k=1},While[!AllTrue[IntegerDigits[k*n],EvenQ],k++];k]; Array[sk,100] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, Mar 27 2015 *)
%o A067044 (PARI) apply( {A067044(n, f=1+n%2)=forstep(a=f*n, oo, f*n, digits(a)%2||return(a/n))}, [1..99]) \\ _M. F. Hasler_, Mar 03 2025
%o A067044 (Python) A067044 = lambda n: next(k for k in range(1+n%2, 9<<99, 1+n%2)if not any(int(d)&1 for d in str(n*k))) # _M. F. Hasler_, Mar 03 2025
%Y A067044 Cf. A061807, A380874.
%Y A067044 Cf. A014263 (numbers with only even digits), A007091 (numbers in base 5).
%K A067044 nonn,base,easy
%O A067044 1,1
%A A067044 _Amarnath Murthy_, Dec 29 2001
%E A067044 More terms from Eli McGowan (ejmcgowa(AT)mail.lakeheadu.ca), May 06 2002
%E A067044 Data corrected by _Paul Tek_, Jun 03 2013