Wikipedia:Reference desk/Archives/Mathematics/2021 July 3

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July 3

Semi-cyclic numbers

Look at the first few pieces of information in the Details section of Cyclic number. It reveals that 076923 is not a cyclic number despite having half of its first few multiples meeting the criterion rather than all that a cyclic number would require. In this case, the answer is yes, no, yes, yes, no, no, no, no, yes, yes, no, yes. (Moreover, all the no's are permutations of the digits 153846.) Are there any other semi-cyclic numbers in base 10?? Georgia guy (talk) 00:44, 3 July 2021 (UTC)

Probably an infinite number. The next is I think 32258064516129. Why? It's the next repeating decimal in this table where the period (n - 1) / 2, where n is the prime numerator of the fraction generating the decimal. n is 31, so the fraction is 1/31. And it looks like it works similarly. The first few products are 32258064516129, 64516129032258, 96774193548387, 129032258064516 – yes, yes, no, yes. After that the next is 1/43, and I imagine they continue after that.2A00:23C8:4588:B01:E12C:4F21:2832:E66C (talk) 05:16, 3 July 2021 (UTC)

See OEIS sequence A097443; for a prime p on this list, the number is n=(10^(p-1)/2-1)/p. Note that p=3 (n=33), is a special case because of repeated digits. The multiples you get correspond to the quadratic residues mod p. OEIS also has similar sequences for a third, a fourth, ... a thirteenth; see the Crossrefs section. All these sequences seem infinite, but I have no idea what the status is on proving it. --RDBury (talk) 08:46, 3 July 2021 (UTC)