[seqfan] Playing with dividing by concatenations

Daniel kimpire at yahoo.com
Mon Apr 9 11:46:03 CEST 2018


Hi all,
I recently added a cross-reference between https://oeis.org/A029455 in which n divides the concatenation of all numbers <=n, and https://oeis.org/A171785, in which n divides the concatenation of all a(1), a(2), ... a(n).
I then decided to look for the next obvious types of concatenation-dividing sequences: those in which n itself is *not* included in the concatenation. For these you'd obviously need to start with defining a(1)=1 and a(2)=2, but the sequence would then go as follows:
   
   - A029455-equivalent, in which n divides the concatenation of all numbers < n: 1, 2, 3, 9, 27...
   - A171785-equivalent, in which n strictly increases and divides the concatenation of all terms a(1) through a(n-1): 1, 2, 3, 41, 43, 129, 9567001, 21147541, 22662659, 23817877,  24837187, 28850377, 28872229, 37916473, 48749751.   


Unless I've made a mistake, I can't find either of these in the database. The closest I found is A240588, in which n divides the concatenation of all terms a(1) through a(n-1) but does *not* strictly increase, rather only defines that n cannot have yet appeared earlier in the sequence. (I will add cross-references to this one. I like cross-references.)

Are these two sequences worth adding? Someone else will have to expand the A029455-equivalent beyond the first few (fairly useless) terms, because the next term requires a concatenation too long for Wolfram Alpha to handle, and I'm just a dabbler so I don't have dedicated mathematics software for doing this sort of thing. But it was able to get quite a few terms of the A171785-equivalent, and can add that one myself.

Daniel Sterman



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