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

1, 11, 2, 25, 50, 27, 500, 7, 4, 2500, 3, 71, 250000, 259, 8, 750000, 10, 39, 5000000, 2598, 7500000000, 77, 9, 6, 2500000000, 5, 53, 533, 75000000001, 38, 383, 43, 75000000000000, 35, 84, 13, 103, 12, 5000000000000, 28, 67, 30, 48, 25000000000000000, 21, 504, 78, 61, 87

Offset: 1

Author

Eric Angelini, Aug 03 2023

Keywords

• base
• nonn

Comments

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.

Examples

			a(1) = 1
a(1) * a(2) = 11
a(1) * a(2) * a(3) = 22
a(1) * a(2) * a(3) * a(4) = 550
a(1) * a(2) * a(3) * a(4) * a(5) = 27500
a(1) * a(2) * a(3) * a(4) * a(5) * a(6) = 742500; etc.
The succession of the above results is:
  1, 11, 22, 550, 27500, 742500, ...
The first terms of the sequence are:
  1, 11, 2, 25, 50, 27, 500, 7, 4, 2500,, ...
We see that the successive digits are the same in the two sequences.
		

Links

• Eric Angelini, Cumulative Sums, Personal blog.

Crossrefs

Cf. A309151, A364664.

Programs

Mathematica

Nest[(a=#;AppendTo[a,(new=Flatten[IntegerDigits/@Table[Times@@a[[;;i]],{i,Length@a}]][[Length@Flatten[IntegerDigits/@a]+1;;]];
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 *)