Narcissistic Number Style Algorithm

Let's consider the following algorithm:

• choose a number \(n\)
• let \(k\) be the number of digits in \(n\)
• raise each digit of \(n\) to the \(k\)-th power and add the results
• call the new number \(n\) and repeat

Let's use \(n=14\) as an example:

• \(14 \rightarrow 1^2 + 4^2 = 17\)
• \(17 \rightarrow 1^2 + 7^2 = 50\)
• \(50 \rightarrow 5^2 + 0^2 = 25\)
• \(25 \rightarrow 2^2 + 5^2 = 29\)
• \(29 \rightarrow 2^2 + 9^2 = 85\)
• \(85 \rightarrow 8^2 + 5^2 = 89\)
• \(89 \rightarrow 8^2 + 9^2 = 145\)
• \(145 \rightarrow 1^3 + 4^3 + 5^3 = 190\)
• \(190 \rightarrow 1^3 + 9^3 + 0^3 = 730\)
• \(730 \rightarrow 7^3 + 3^3 + 0^3 = 370\)
• \(370 \rightarrow 3^3 + 7^3 + 0^3 = 370\) 

370 is a narcissistic number as explained in my post Narcissistic, D-Powerfull and Friedman Numbers. The trajectory of any number under this algorithm will either end with a narcissistic number (as was the case with 14) or it will enter a loop (as is the case with 28218). The latter has the following trajectory (permalink):

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TRAJECTORY ANALYSIS FOR INPUT: 28218

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Full Trajectory Visited:

28218, 65601, 18678, 90120, 59082, 94974, 136953, 595181, 824837, 646826, 406272, 168529, 855931, 825565, 355739, 681798, 1220035, 80569, 102718, 379859, 1459029, 9660576, 6524445, 485466, 379273, 768261, 473170, 240124, 8321, 4194, 7074, 5058, 5346, 2258, 4753, 3363, 1539, 7268, 7809, 13058, 36137, 25070, 19964, 126899, 1371747, 2489202, 6896889, 16417266, 10869443, 61641187, 25966788, 86116067, 27580867, 47154531, 6683686, 5316235, 440689, 848433, 533938, 811397, 911965, 1125165, 436317, 169860, 886898, 1626673, 1665667, 2021413, 18829, 124618, 312962, 578955, 958109, 1340652, 376761, 329340, 537059, 681069 -> [loops back to 886898]

Loop Entry Point:   886898 (encountered at step 65)

Pre-period Length:  64 step(s) before entering cycle

Cycle Length:       14 distinct number(s) in the loop

Canonical Cycle:    18829, 124618, 312962, 578955, 958109, 1340652, 376761, 329340, 537059, 681069, 886898, 1626673, 1665667, 2021413

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Figure 1 shows a graph of its trajectory:

Figure 1: permalink

I've incorporated this algorithm into my daily number analysis.