A narcissistic number is a natural number which is equal to the sum of its digits when each digit is taken to the power of the number digits. For example \$8208 = 8^4 + 2^4 + 0^4 + 8^4\$, so is narcissistic.
We'll define a function \$f(x)\$ as the following, for a natural number \$x = d_1d_2\dots d_n\$, where \$d_i\$ is a single digit \$0\$ to \$9\$ (therefore \$x\$ has \$n\$ digits):
$$f(x) = \sum_{i=1}^nd_i^n$$
In this case, a number is narcissistic if \$f(x) = x\$.
However, when we apply \$f(x)\$ to a number repeatedly, we find an interesting pattern emerges. For any given \$x\$, the sequence either reaches a fixed point (i.e. a narcissistic number), or enters a fixed loop which repeats infinitely. For examples, take the three integers \$x = 104, 127, 370\$:
\$x = 104\$: Repeated application of \$f(x)\$ leads to the following chain
$$104, 65, 61, 37, 58, 89, 145, 190, 730, 370, 370, ...$$
Here, the loop eventually reaches a fixed point, \$370\$.
\$x = 127\$: Repeated application of \$f(x)\$ leads to
$$127, 352, 160, 217, 352, 160, 217, ...$$
Here, the triple \$352, 160, 217\$ repeats ad infinitum
\$x = 370\$: \$x\$ here is already narcissistic, so the chain will just be an endless stream of \$370\$s.
These examples document the two possible outcomes for a given \$x\$. By treating a fixed point as a loop of length \$1\$, we now arrive at the task at hand:
Given a natural number \$n > 0\$, output the length of the loop that arises through repeated application of \$f(n)\$.
The above three examples return \$1\$, \$3\$ and \$1\$ respectively. You can assume that all \$n\$ eventually enter a loop, which appears to be the case for at least all \$n < 10^4\$. If there exists an \$n\$ for which this is false, your program may do anything short of summoning Cthulhu.
This is code-golf so the shortest code in bytes wins.
The vast majority of numbers return \$1\$. However, these two arrays contain all \$n < 1000\$ which don't, along with what their outputs should be:
[ 59, 95, 106, 115, 127, 136, 138, 147, 149, 151, 157, 159, 160, 163, 168, 169, 172, 174, 175, 177, 178, 179, 183, 186, 187, 189, 194, 195, 196, 197, 198, 199, 217, 228, 229, 235, 238, 244, 245, 253, 254, 255, 258, 259, 267, 268, 271, 276, 277, 279, 282, 283, 285, 286, 289, 292, 295, 297, 298, 299, 309, 316, 318, 325, 328, 335, 352, 353, 355, 357, 358, 361, 366, 367, 369, 375, 376, 381, 382, 385, 388, 389, 390, 396, 398, 405, 408, 417, 419, 424, 425, 442, 445, 447, 450, 452, 454, 456, 457, 459, 465, 466, 468, 469, 471, 474, 475, 477, 478, 479, 480, 486, 487, 488, 491, 495, 496, 497, 499, 504, 507, 508, 511, 517, 519, 523, 524, 525, 528, 529, 532, 533, 535, 537, 538, 540, 542, 544, 546, 547, 549, 552, 553, 555, 556, 558, 559, 564, 565, 567, 568, 570, 571, 573, 574, 576, 580, 582, 583, 585, 586, 589, 591, 592, 594, 595, 598, 601, 607, 609, 610, 613, 618, 619, 627, 628, 631, 636, 637, 639, 645, 646, 648, 649, 654, 655, 657, 658, 663, 664, 666, 669, 670, 672, 673, 675, 678, 679, 681, 682, 684, 685, 687, 689, 690, 691, 693, 694, 696, 697, 698, 699, 705, 706, 708, 712, 714, 715, 717, 718, 719, 721, 726, 727, 729, 735, 736, 741, 744, 745, 747, 748, 749, 750, 751, 753, 754, 756, 760, 762, 763, 765, 768, 769, 771, 772, 774, 777, 778, 779, 780, 781, 784, 786, 787, 788, 791, 792, 794, 796, 797, 799, 804, 805, 807, 813, 816, 817, 819, 822, 823, 825, 826, 829, 831, 832, 835, 838, 839, 840, 846, 847, 848, 850, 852, 853, 855, 856, 859, 861, 862, 864, 865, 867, 869, 870, 871, 874, 876, 877, 878, 883, 884, 887, 891, 892, 893, 895, 896, 900, 903, 906, 914, 915, 916, 917, 918, 919, 922, 925, 927, 928, 929, 930, 936, 938, 941, 945, 946, 947, 949, 951, 952, 954, 955, 958, 960, 961, 963, 964, 966, 967, 968, 969, 971, 972, 974, 976, 977, 979, 981, 982, 983, 985, 986, 991, 992, 994, 996, 997, 999]
[ 3, 3, 3, 3, 3, 2, 10, 14, 10, 3, 10, 14, 3, 2, 14, 10, 3, 14, 10, 2, 10, 2, 10, 14, 10, 10, 10, 14, 10, 2, 10, 10, 3, 10, 3, 3, 3, 2, 2, 3, 2, 10, 10, 10, 14, 10, 3, 14, 10, 14, 10, 3, 10, 10, 10, 3, 10, 14, 10, 10, 14, 2, 10, 3, 3, 2, 3, 2, 10, 10, 10, 2, 10, 10, 14, 10, 10, 10, 3, 10, 14, 6, 14, 14, 6, 10, 14, 14, 10, 2, 2, 2, 3, 14, 10, 2, 3, 10, 3, 10, 10, 10, 14, 10, 14, 14, 3, 14, 10, 10, 14, 14, 10, 10, 10, 10, 10, 10, 10, 10, 14, 10, 3, 10, 14, 3, 2, 10, 10, 10, 3, 2, 10, 10, 10, 10, 2, 3, 10, 3, 10, 10, 10, 10, 10, 14, 3, 10, 10, 14, 10, 14, 10, 10, 3, 14, 10, 10, 10, 14, 10, 10, 14, 10, 10, 3, 10, 3, 3, 10, 3, 2, 14, 10, 14, 10, 2, 10, 10, 14, 10, 10, 14, 10, 10, 10, 14, 10, 10, 10, 14, 10, 3, 14, 10, 14, 2, 10, 14, 10, 14, 10, 2, 6, 10, 10, 14, 10, 10, 10, 6, 2, 14, 3, 14, 3, 14, 10, 2, 10, 2, 3, 14, 10, 14, 10, 10, 14, 14, 3, 14, 10, 10, 14, 10, 10, 3, 14, 3, 14, 10, 14, 2, 10, 2, 10, 14, 6, 14, 14, 14, 10, 10, 2, 14, 14, 2, 14, 10, 10, 14, 3, 14, 10, 14, 10, 14, 10, 10, 10, 3, 10, 10, 10, 10, 3, 10, 14, 6, 14, 14, 10, 10, 10, 10, 10, 14, 10, 10, 14, 10, 14, 10, 2, 6, 14, 10, 10, 2, 14, 14, 14, 10, 14, 10, 10, 6, 10, 6, 14, 14, 10, 10, 14, 10, 2, 10, 10, 3, 10, 14, 10, 10, 14, 14, 6, 10, 10, 10, 10, 10, 14, 10, 10, 3, 10, 10, 10, 14, 10, 10, 10, 6, 2, 2, 14, 10, 10, 14, 3, 10, 10, 6, 10, 6, 10, 10, 10, 2, 3, 2]
Furthermore, this is a program which takes an integer \$n\$ and, for each integer \$1 \le i \le n\$, generates it's output and the loop that arises.
This is a question over on Math.SE about whether the output ever exceeds \$14\$, and if each natural number will eventually go into a loop or not.