A343336 -id:A343336 - cp's OEIS frontend

A110303 Alternators.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 1

Walter Nissen, Jul 18 2005

An alternating integer is a positive integer for which, in base-10, the parity of its digits alternates. E.g., 121 is alternating because its consecutive digits are odd-even-odd, 1 being odd and 2 even. Of course, 1234567890 is also alternating. An alternator is a positive integer which has a multiple which is alternating.

This sequence is the answer to the 6th problem proposed the 2nd day by Iran during the 45th International Mathematical Olympiad, in Athens (Greece), 2004 (see links). - Bernard Schott, Apr 12 2021

			11 is an alternator and in the sequence because it has a multiple which is alternating. The least of these multiples is 121.

Michael De Vlieger, Table of n, a(n) for n = 1..10001 (adapted to offset by Michel Marcus)
45th International Mathematical Olympiad (45th IMO), Problem #6 and Solution, Mathematics Magazine, 2005, Vol. 78, No. 3, pp. 247, 250-251.
The IMO Compendium, Problem 6, 45th IMO 2004.
Index to sequences related to Olympiads.

Cf. A030141, A030142, A110304, A110305, A008602 (complement), A343335, A343336.

Select[Range[75], Mod[#, 20] != 0 &] (* Michael De Vlieger, Apr 13 2021 *)

Positive n, not congruent to 0 mod 20.

a(n + 19) = a(n) + 20. - David A. Corneth, Apr 13 2021

Offset 1 from Michel Marcus, May 12 2021

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.

Original entry on oeis.org

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

Bernard Schott, Apr 12 2021

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.

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

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

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

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 *)

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A110303 Alternators.

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