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A364697 Lexicographically earliest permutation of the positive integers such that the successive cumulative products reproduce the sequence itself, digit by digit.

%I A364697 #15 Aug 05 2023 12:59:47
%S A364697 1,11,2,25,50,27,500,7,4,2500,3,71,250000,259,8,750000,10,39,5000000,
%T A364697 2598,7500000000,77,9,6,2500000000,5,53,533,75000000001,38,383,43,
%U A364697 75000000000000,35,84,13,103,12,5000000000000,28,67,30,48,25000000000000000,21,504,78,61,87
%N A364697 Lexicographically earliest permutation of the positive integers such that the successive cumulative products reproduce the sequence itself, digit by digit.
%C A364697 If we want the sequence to be the lexicographically earliest permutation of the integers > 0, we must start with a(1) = 1 and a(2) = 11. With a(2) < 11, the sequence stops immediately.
%H A364697 Eric Angelini, <a href="http://cinquantesignes.blogspot.com/2023/08/cumulative-sums.html">Cumulative Sums</a>, Personal blog.
%e A364697 a(1) = 1
%e A364697 a(1) * a(2) = 11
%e A364697 a(1) * a(2) * a(3) = 22
%e A364697 a(1) * a(2) * a(3) * a(4) = 550
%e A364697 a(1) * a(2) * a(3) * a(4) * a(5) = 27500
%e A364697 a(1) * a(2) * a(3) * a(4) * a(5) * a(6) = 742500; etc.
%e A364697 The succession of the above results is:
%e A364697   1, 11, 22, 550, 27500, 742500, ...
%e A364697 The first terms of the sequence are:
%e A364697   1, 11, 2, 25, 50, 27, 500, 7, 4, 2500,, ...
%e A364697 We see that the successive digits are the same in the two sequences.
%t A364697 Nest[(a=#;AppendTo[a,(new=Flatten[IntegerDigits/@Table[Times@@a[[;;i]],{i,Length@a}]][[Length@Flatten[IntegerDigits/@a]+1;;]];
%t A364697 k=1;While[MemberQ[a,FromDigits@new[[;;k]]]||new[[k+1]]==0,k++];FromDigits@new[[;;k]])])&,{1,11,2,25},45] (* _Giorgos Kalogeropoulos_, Aug 05 2023 *)
%Y A364697 Cf. A309151, A364664.
%K A364697 base,nonn
%O A364697 1,2
%A A364697 _Eric Angelini_, Aug 03 2023