## Monday, April 08, 2024

### A Falcoz digit-fancy

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In addition to my own "fanciful extension" of Éric Angelini's Two identical digits effort, Jean-Marc Falcoz suggested his own variation (at the end of the blog entry): "Lexicographically earliest sequence of distinct positive terms such that [the product of adjacent terms] contains exactly 1 digit 1 (if 1 is present), 2 digits 2 (if 2 is present), 3 digits 3 (if 3 is present), ... 9 digits 9 (if 9 is present)." He presented 113 terms of the sequence but I was hungry for more.

My initial plot (above) just exceeds 1000 terms. My updated (May 13) plot (below) extends this to 5000 terms. Term #4367 = 1785221551, a local maximum. Term #2213 = 2. Also known to appear are 8, 11, 13, 17, 22, 44, 84, 97, ... Possible products are A108571. Our indexed products are such that product #2 is term #2 multiplied by term #1 (product #1 is 1 by fiat). In the current product list we have 16 duplicates and 2 triplicates (#3 = #2215 = #2219 and #416 = #2349 = #3632). If typed by their constituent digits, regardless of digit order, the number of possible types is given by A125573. Our current list realizes 105 of these, the number of which (sorted by product digit-length) are: 1, 1, 1, 2, 1, 3, 4, 4, 6, 7, 8, 11, 10, 14, 13, 11, 7, 1.

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