A343336 Factors of alternators which produce least alternating proper multiples.

2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 11, 3, 4, 4, 2, 2, 2, 2, 2, 0, 3, 19, 3, 3, 2, 2, 2, 2, 2, 3, 11, 3, 5, 7, 2, 2, 2, 2, 2, 0, 3, 5, 3, 14, 2, 2, 2, 2, 2, 5, 11, 8, 4, 4, 3, 11, 8, 4, 4, 0, 3, 7, 3, 4, 5, 13, 10, 4, 4, 3, 11, 3, 4, 4, 6, 6, 5, 3, 6, 0, 7, 5, 6, 3, 9, 3, 8, 7, 10

Offset: 1

Author

Bernard Schott, Apr 15 2021

Keywords

• nonn
• base

Comments

Every positive integer that is not multiple of 20 is called an alternator (A110303) because it has a multiple in which parity of the decimal digits alternates and that is called an alternating integer (A030141).

If n is an alternator, n <> 20*k, a(n) is the smallest q > 1, such that q*n is a proper alternating multiple of n; this is a variant of A110305 where q = 1 is authorized when n is an alternating alternator.

If n is congruent to 0 mod 20, a(n) is set to zero to indicate that n is not an alternator.

Examples

			a(14) = 4 because the successive proper multiples of 14 are 28, 42 that are not alternating, then, 4*14 = 56 is alternating because 5 is odd and 6 is even.
		

Links

• The IMO Compendium, Problem 6, 45th IMO 2004.

Crossrefs

Cf. A030141, A110303, A110304, A110305, A343335.

Programs

Maple

f:= proc(n) local k,L;
  if n mod 20 = 0 then return 0 fi;
  if n <= 4 then return 2 fi;
  for k from 2 do
    L:= convert(k*n,base,10) mod 2;
    if convert(L[1..-2]+L[2..-1],set) = {1} then return k fi;
  od
end proc:
map(f, [$1..100]); # Robert Israel, Apr 15 2021

Mathematica

altQ[n_] := (r = Mod[IntegerDigits[n], 2]) == Split[r, UnsameQ][[1]]; a[n_] := If[Divisible[n, 20], 0, Module[{k = 2*n}, While[! altQ[k], k += n]; k/n]]; Array[a, 100] (* Amiram Eldar, Apr 15 2021 *)

Formula

a(n) >= A110305(n).

Extensions

Name edited by Michel Marcus, May 12 2021