More About 366

My previous post was about Compositions of 365 and 366 but just tonight, while playing around with numbers, I came across an interesting property of the number 366. It started out by considering a variation on my odds and evens algorithm that I've written about extensively in previous posts. In that algorithm, starting with any number, the odd digits of the number are added to it and the even digits subtracted to form a new number (unless the odd and even digits cancel out). This process is continued with the new number until a fixed point is reached or a loop is entered.

I thought about what would happen if the product of the digits of a number were added to the number to form a new number and the process repeated until some sort of resolution was reached. The process will clearly terminate once a zero digit appears. I wondered how many repetitions might be required. My investigations of the first 1000 numbers revealed that the maximum number of repetitions was 27 and this occurred with three numbers: 187, 248 and 264.

Here are their trajectories:

187, 243, 267, 351, 366, 474, 586, 826, 922, 958, 1318, 1342, 1366, 1474, 1586, 1826, 1922, 1958, 2318, 2366, 2582, 2742, 2854, 3174, 3258, 3498, 4362, 4506

248, 312, 318, 342, 366, 474, 586, 826, 922, 958, 1318, 1342, 1366, 1474, 1586, 1826, 1922, 1958, 2318, 2366, 2582, 2742, 2854, 3174, 3258, 3498, 4362, 4506

264, 312, 318, 342, 366, 474, 586, 826, 922, 958, 1318, 1342, 1366, 1474, 1586, 1826, 1922, 1958, 2318, 2366, 2582, 2742, 2854, 3174, 3258, 3498, 4362, 4506

Each sequence consists of 28 members and the members of the sequences are identical from 366 onwards. The sequences beginning with 264 and 248 differ only in the initial term but the trajectory of 187 does not meet up with them until 366 and the first four terms are odd (187, 243, 267, 351). After 23 steps, all leading to even numbers, 366 reaches a dead end.

Thus 366 forms a sequence consisting of 24 terms:

366, 474, 586, 826, 922, 958, 1318, 1342, 1366, 1474, 1586, 1826, 1922, 1958, 2318, 2366, 2582, 2742, 2854, 3174, 3258, 3498, 4362, 4506

366 is the first number to generate a sequence of length 24. Up to 1,000,000, the numbers that produce a sequence of this length are:

366, 393, 426, 442, 24567, 24627, 124567, 124627, 158637, 174219, 184441, 273867, 419976, 441136, 441232, 513411, 581746 

Six of the resulting sequences are subsets of the sequences containing 28 terms, described earlier. Getting back to the sequences of maximal length, the numbers up to ten million that lead to sequences of length 28 are:

187, 248, 264, 386776, 874423, 1386776, 3169526, 3175571, 3241862, 3795455, 3829475, 3843299, 4213657, 4241417, 4293567, 4322139, 4446651, 4449531, 4452571, 5113891, 5114811

Thus 187, 243, 248, 264, 267, 312, 318, 342, 351 all lead (directly or indirectly) to 366 on the way to 4506, under the add product of digits to number recursive process. See Figure 1.

Figure 1

Why the maximum length of the sequences should be 28 (consisting of 27 steps leading to a dead end) remains a mystery to me. There's clearly something of significance in the number 27 and maybe I'll dig deeper at a later date.