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a(n) is the smallest proper alternating multiple of n when n is not a multiple of 20, otherwise a(20*k) = 0 for k >= 1.
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%I #20 May 10 2021 09:25:27

%S 2,4,6,8,10,12,14,16,18,30,121,36,52,56,30,32,34,36,38,0,63,418,69,72,

%T 50,52,54,56,58,90,341,96,165,238,70,72,74,76,78,0,123,210,129,616,90,

%U 92,94,96,98,250,561,416,212,216,165,616,456,232,236,0,183,434,189,256,325,858

%N a(n) is the smallest proper alternating multiple of n when n is not a multiple of 20, otherwise a(20*k) = 0 for k >= 1.

%C 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 (see link) and that is called an alternating integer (A030141).

%C If n is an alternator, n <> 20*k, k>=1, then a(n) is the smallest alternating multiple k*n, with k > 1.

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

%C This sequence is a variant of A110304, but here the smallest alternating multiple of n cannot be n, when n is an alternating integer.

%H The IMO Compendium, <a href="https://imomath.com/othercomp/I/Imo2004.pdf">Problem 6</a>, 45th IMO 2004.

%e For n = 13, 2 * 13 = 26, 3 * 13 = 39, 4 * 13 = 52 that is alternating, so, a(13) = 52.

%t 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]]; Array[a, 100] (* _Amiram Eldar_, Apr 12 2021 *)

%Y Cf. A030141, A110303, A110304, A110305, A343336.

%K nonn,base

%O 1,1

%A _Bernard Schott_, Apr 12 2021