Sam's New Sequence

"Sam" is actually an anagram of the acronym for the rule: S eparate, M ultiply, A dd. The proper name for these sequences is Digit Product Sequences .

$1:$ Separate the term into its digits (i.e. $45$ becomes $4, 5$ ).

$2:$ Multiply the non-zero digits together (i.e. $45\to(4,5)\to4*5=20$ ).

$3:$ Add the resulting product of digits to the original term (i.e. $20+45=65$ ).

In a nutshell, add the product of the digits of $a_{n-1}$ to $a_{n-1}$ to obtain $a_n$ .

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Note: Notice that when we start with a seed value $a_0=3$ , we obtain the sequence

$\large 3, 6, 14, 18, 26...$

This sequence 'meets up' with the sequence started by the seed value $A_0 = 1$ at $a_4$ and $A_7$ . It has been conjectured that this 'meeting up' behavior of the Digit Product Sequence occurs for every pair of seed values.

Credit : Paul Loomis, An Interesting Family of Iterated Sequences