A067044 Smallest positive k such that k*n contains only even digits.
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, 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, 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
Offset: 1
Examples
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. 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
Links
- Paul Tek, Table of n, a(n) for n = 1..10000
Crossrefs
Programs
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Mathematica
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 *) 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 *) -
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 -
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
Formula
From M. F. Hasler, Mar 07 2025: (Start)
There is an explicit formula for many values of n:
a(n) = 1 if n has only even digits <=> n is in A014263, else:
a(n) = 2 if n has only digits < 5 <=> n is in A007091;
a(m*(10^k-1)) = 8*round(10^k/6)^2/m for m = 1, 2, 4 or 8 and any k > 0;
a(5*(10^k-1)) = 16*round(10^k/6)^2 for any k > 0;
a(50*m + {5 or 15}) = 4 if m has all digits < 5. (End)
Extensions
More terms from Eli McGowan (ejmcgowa(AT)mail.lakeheadu.ca), May 06 2002
Data corrected by Paul Tek, Jun 03 2013
Comments