A110734 Let n = a_1a_2...a_k, where the a_i are digits. a(n) = least multiple of n of the form a_1b_1a_2b_2...b_{k-1}a_k, obtained by inserting single digits b_i in the gaps; 0 if no such number exists.

1, 2, 3, 4, 5, 6, 7, 8, 9, 100, 121, 132, 143, 154, 105, 176, 187, 108, 0, 200, 231, 242, 253, 264, 225, 286, 297, 0, 0, 300, 341, 352, 363, 374, 315, 396, 0, 0, 0, 400, 451, 462, 473, 484, 405, 0, 0, 0, 0, 500, 561, 572, 583, 594, 0, 0, 0, 0, 0, 600, 671, 682, 693, 0, 0

Offset

1,2

Comments

Conjecture: for large n, a(n) is nonzero. For a k-digit number there are k-1 gaps and 10^(k-1) candidates, so the chances that one of them is a multiple of n increases with k on the one hand though the probability decreases because n becomes large.

Pursuing the probability argument, the probability that a(n) is zero is (1-1/n)^{10^{k-1}}, which has an expected value of e^{-(10^{k-1})/n}. (10^{k-1})/n varies from 1/100 to 1/10, depending on the leading digits of n, so the probability a(n) is zero is between e^{-1/10} and e^{-1/100}; thus we would expect that only a small but nonzero fraction of all n have a(n) nonzero. Of course, it is not clear that the probability argument is accurate. Franklin T. Adams-Watters, Sep 25 2006

Links

Table of n, a(n) for n=1..65.

Crossrefs

Cf. A080436, A110735.

Sequence in context: A088472 A087019 A044911 * A080436 A135295 A122625

Adjacent sequences: A110731 A110732 A110733 * A110735 A110736 A110737

Keyword

base,easy,nonn

Author

Amarnath Murthy, Aug 09 2005

Extensions

Edited, corrected and extended by Franklin T. Adams-Watters, Sep 25 2006

Status

approved