login
Smallest precursor of n-th cycle in the "Recurring Digital Invariant Variant" problem.
4

%I #6 Jun 09 2016 13:34:27

%S 1,2,3,4,5,6,7,8,9,14,59,108,119,136,138,147,177,389,407,559,709,999,

%T 1118,1157,1346,4479,11227,12399,22779,30489,100666,127779,577999,

%U 677779,1000259,1001458,1007889,1035889,1124577,1188888

%N Smallest precursor of n-th cycle in the "Recurring Digital Invariant Variant" problem.

%C The problem is the following:

%C a) choose a number N

%C b) let k be the number of digits in N

%C c) raise each digit of N to the k-th power and add the results

%C d) call the new number N and repeat

%C Example:

%C a) 14 = N

%C b) k = 2

%C c) 1^2 + 4^2 = 17

%C d) 17 = N

%C e) k = 2

%C f) 1^2 + 7^2 = 50

%C g) 50 = N

%C ... etc.

%C Here is the trajectory of 14:

%C 14 -> 1^2 + 4^2 = 17

%C 17 -> 1^2 + 7^2 = 50

%C 50 -> 5^2 + 0^2 = 25

%C 25 -> 2^2 + 5^2 = 29

%C 29 -> 2^2 + 9^2 = 85

%C 85 -> 8^2 + 5^2 = 89

%C 89 -> 8^2 + 9^2 = 145

%C 145 -> 1^3 + 4^3 + 5^3 = 190

%C 190 -> 1^3 + 9^3 + 0^3 = 730

%C 730 -> 7^3 + 3^3 + 0^3 = 370

%C 370 -> 3^3 + 7^3 + 0^3 = 370 (fixed point)

%C The question is, what are the cycles that appear in the trajectories?

%C The following table of the first 34 cycles (arranged in order of the smallest precursor) was calculated by Hans Havermann:

%C The format for each cycle is:

%C Index {the smallest precursor (the current sequence), the cycle length, {the cycle itself with the smallest element of the cycle first - see A151544}}:

%C 1 { 1, 1, {1}}

%C 2 { 2, 1, {2}}

%C 3 { 3, 1, {3}}

%C 4 { 4, 1, {4}}

%C 5 { 5, 1, {5}}

%C 6 { 6, 1, {6}}

%C 7 { 7, 1, {7}}

%C 8 { 8, 1, {8}}

%C 9 { 9, 1, {9}}

%C 10 { 14, 1, {370}}

%C 11 { 59, 3, {160, 217, 352}}

%C 12 { 108, 1, {153}}

%C 13 { 119, 1, {371}}

%C 14 { 136, 2, {136, 244}}

%C 15 { 138, 10, {259, 862, 736, 586, 853, 664, 496, 1009, 6562, 3233}}

%C 16 { 147, 14, {18829, 124618, 312962, 578955, 958109, 1340652, 376761, 329340, 537059, 681069, 886898, 1626673, 1665667, 2021413}}

%C 17 { 177, 2, {58618, 76438}}

%C 18 { 389, 6, {2929, 13154, 4394, 7154, 3283, 4274}}

%C 19 { 407, 1, {407}}

%C 20 { 559, 3, {282595, 824963, 845130}}

%C 21 { 709, 1, {8208}}

%C 22 { 999, 2, {2178, 6514}}

%C 23 { 1118, 4, {10933, 59536, 73318, 50062}}

%C 24 { 1157, 12, {5908997, 17347727, 23131558, 17571846, 30442597, 49340036, 44870531, 23070276, 13216291, 44733413, 5981093, 11743403}}

%C 25 { 1346, 1, {1634}}

%C 26 { 4479, 1, {9474}}

%C 27 { 11227, 1, {54748}}

%C 28 { 12399, 1, {32164049651}}

%C 29 { 22779, 1, {92727}}

%C 30 { 30489, 1, {93084}}

%C 31 {100666, 12, {1680387, 5299971, 15250704, 6611844, 2689794, 12783081, 39326052, 45130596, 45579685, 68505765, 27073124, 11602212}}

%C 32 {127779, 1, {548834}}

%C 33 {577999, 1, {4210818}}

%C 34 {677779, 3, {2767918, 8807272, 5841646}}

%C 35 {1000259, 1, {9926315}}

%C 36 {1001458, 6, {2191663, 5345158, 2350099, 9646378, 8282107, 5018104}}

%C 37 {1007889, 1, {9800817}}

%C 38 {1035889, 2, {8139850, 9057586}}

%C 39 {1124577, 1, {1741725}}

%C 40 {1188888, 1, {24678051}}

%C 41 {2055779, 2, {2755907, 6586433}}

%C 42 {2566699, 1, {472335975}}

%C 43 {4888888, 10, {180450907, 564207094, 440329717, 468672187, 369560719, 837322786, 359260756, 451855933, 527799103, 857521513}}

%C 44 {10135679, 1, {24678050}}

%C 45 {10146899, 1, {146511208}}

%C 46 {10233389, 1, {88593477}}

%C 47 {10266888, 7, {1139785743, 5136409024, 3559173428, 4863700423, 1418899523, 9131926726, 7377037502}}

%C 48 {14489999, 3, {180975193, 951385123, 525584347}}

%C 49 {14788889, 1, {912985153}}

%C 50 {20248999, 1, {534494836}}

%C 51 {155999999, 2, {277668893, 756738746}}

%C Any number < 10^9 will fall into one of these 51 cycles.

%C The name "Recurring Digital Invariant Variant" was suggested by Mensanator on the rec.puzzles web site.

%H Hans Havermann, <a href="/A151543/b151543.txt">Table of n, a(n) for n = 1..51</a>

%H Eric Angelini, <a href="http://www.cetteadressecomportecinquantesignes.com/RecurDigit.htm">A Recurring Digital Invariant Variant</a>

%H E. Angelini, <a href="/A151543/a151543.pdf">A Recurring Digital Invariant Variant</a> [Cached, with permission]

%Y Cf. A005188, A151544, A157714.

%K nonn,base

%O 1,2

%A _N. J. A. Sloane_, May 15 2009 based on email from _Eric Angelini_, Feb 18 2009