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%I #15 Jan 15 2019 20:31:33
%S 0,902,29022,4290222,342902224,13429022244,6134290222446,
%T 161342902224468,31613429022244686,5316134290222446864,
%U 753161342902224468642,17531613429022244686428,6175316134290222446864283,561753161342902224468642837,35617531613429022244686428373,8356175316134290222446864283739
%N Lexicographically earliest infinite sequence of envelope numbers that are inserted into the largest possible envelopes (see the Comments and the Crossrefs sections).
%C Every term is equal to the term after it, except for the first and the last digit; those two digits, concatenated in that order, constitute the envelope; a(n) is always a multiple of a(n+1)'s envelope, this envelope being the largest possible so that the sequence doesn't stop.
%H Jean-Marc Falcoz, <a href="/A323143/b323143.txt">Table of n, a(n) for n = 1..135</a>
%e The first term > 0 is 902, and we see that 902 is indeed an envelope number as 0 is a multiple of 92; as we want always to insert an envelope number into the biggest next possible envelope, we try first 909, but 909 has no 2-digit divisor; nor could we select 908 (for the same reason), nor 907 (as 907 is prime), nor 906 (as 906 too has no 2-digit divisor), nor 905 or 904 (for the same reason), nor 903 (as 903 would produce the envelope number 29031 which has itself no 2-digit divisor); 902 is ok as 902 has two 2-digit divisors, 11 and 22: we thus keep the biggest one (22) to build a(3) = 29022; a(3) has itself three 2-digit divisors, 14, 21 and 42: we keep the biggest one to build a(4) = 4290222; this number has 16 divisors altogether, but three of them are 2-digit numbers, 17, 34 and 51; we will keep 34 (as 51 cannot produce an infinite sequence) to build a(5) = 342902224, etc.
%e A way to spare some space in presenting this sequence, would be to align, after a(1) = 0, the successive divisors that are kept; this would form the sequence 0,92,22,42,34,14,66,18,36,54,72,18,63,57,33,89,47,11,21,43,59,47,79,67,61,43,27,27,99,81,19,17,63,27,99,27,27,93,71,59,11,49,11,69,87,39,79,79,91,33,39,43,53,43,11,93,27,51,57,81,81,97,29,99,91,37,37,21,33,69,63,33,39,23,61,51,27,33,23,73,11,67,99,27,27,69,...
%Y Cf. A323142 (the definition of an envelope number and the first simple ones).
%K base,nonn
%O 1,2
%A _Eric Angelini_ and _Jean-Marc Falcoz_, Jan 10 2019