A343335 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.

2, 4, 6, 8, 10, 12, 14, 16, 18, 30, 121, 36, 52, 56, 30, 32, 34, 36, 38, 0, 63, 418, 69, 72, 50, 52, 54, 56, 58, 90, 341, 96, 165, 238, 70, 72, 74, 76, 78, 0, 123, 210, 129, 616, 90, 92, 94, 96, 98, 250, 561, 416, 212, 216, 165, 616, 456, 232, 236, 0, 183, 434, 189, 256, 325, 858

Offset

1,1

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

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

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

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

Links

Table of n, a(n) for n=1..66.

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

Example

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

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]]; Array[a, 100] (* Amiram Eldar, Apr 12 2021 *)

Crossrefs

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

Sequence in context: A066122 A119766 A030143 * A110725 A100800 A169921

Adjacent sequences: A343332 A343333 A343334 * A343336 A343337 A343338

Keyword

nonn,base

Author

Bernard Schott, Apr 12 2021

Status

approved