A374725 The "multiplicative comma sequence": the lexicographically earliest sequence of positive integers with the property that the sequence formed by the pairs of digits adjacent to the commas between the terms is the same as the sequence of successive ratios between the terms.

1, 11, 121, 1331, 14641, 161051, 1771561, 19487171, 233846052, 5846151300

Offset: 1

Author

Nicholas M. R. Frieler, Jul 17 2024

Keywords

• nonn
• base
• fini
• full

Comments

A more formal definition can be given as follows: a(1) = 1; for n > 1, let x be the least significant digit of a(n-1); then a(n) = a(n-1) * (10*x + y), with y being the most significant digit of a(n). Choose the smallest such y if such a y exists. If no such y exists, the sequence ends. We also restrict y to being a nonzero digit.

The sequence is given in its entirety as there is no possible next term after 5846151300.

Choosing other values for a(1) yields finite sequences up to a(1) = 10000 as long as a(1) is not of the form 1...0 otherwise the sequence is constant and infinite. For example, if a(1) = 120, then a(2) = 120 because 120 * 01 = 120.

Examples

			Replace each comma in the original sequence by the pair of digits adjacent to the comma; the result is the sequence of first ratios between the terms of the sequence:
Sequence: 1, 11, 121, 1331, 14641, 161051, 1771561, 19487171, 233846052, 5846151300
Ratios:    11, 11,  11,   11,    11,     11,      11,       12,        25
For example: a(9) = 233846052 = 12 * 19487171 = 12 * a(8)
		

Crossrefs

Cf. A121805.

Programs

Mathematica

a[1] = 1; a[n_] := a[n] = For[x = Mod[a[n - 1], 10]; y = 1, y <= 9, y++, an = a[n - 1]*(10*x + y); If[y == IntegerDigits[an][[1]], Return[an]]]; Array[a, 10]