A151543 Smallest precursor of n-th cycle in the "Recurring Digital Invariant Variant" problem.

1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 59, 108, 119, 136, 138, 147, 177, 389, 407, 559, 709, 999, 1118, 1157, 1346, 4479, 11227, 12399, 22779, 30489, 100666, 127779, 577999, 677779, 1000259, 1001458, 1007889, 1035889, 1124577, 1188888

Offset: 1

Author

N. J. A. Sloane, May 15 2009 based on email from Eric Angelini, Feb 18 2009

Keywords

• nonn
• base

Comments

The problem is the following:

a) choose a number N

b) let k be the number of digits in N

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

d) call the new number N and repeat

Example:

a) 14 = N

b) k = 2

c) 1^2 + 4^2 = 17

d) 17 = N

e) k = 2

f) 1^2 + 7^2 = 50

g) 50 = N

... etc.

Here is the trajectory of 14:

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

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

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

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

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

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

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

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

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

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

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

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

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

The format for each cycle is:

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

1 { 1, 1, {1}}

2 { 2, 1, {2}}

3 { 3, 1, {3}}

4 { 4, 1, {4}}

5 { 5, 1, {5}}

6 { 6, 1, {6}}

7 { 7, 1, {7}}

8 { 8, 1, {8}}

9 { 9, 1, {9}}

10 { 14, 1, {370}}

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

12 { 108, 1, {153}}

13 { 119, 1, {371}}

14 { 136, 2, {136, 244}}

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

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

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

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

19 { 407, 1, {407}}

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

21 { 709, 1, {8208}}

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

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

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

25 { 1346, 1, {1634}}

26 { 4479, 1, {9474}}

27 { 11227, 1, {54748}}

28 { 12399, 1, {32164049651}}

29 { 22779, 1, {92727}}

30 { 30489, 1, {93084}}

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

32 {127779, 1, {548834}}

33 {577999, 1, {4210818}}

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

35 {1000259, 1, {9926315}}

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

37 {1007889, 1, {9800817}}

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

39 {1124577, 1, {1741725}}

40 {1188888, 1, {24678051}}

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

42 {2566699, 1, {472335975}}

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

44 {10135679, 1, {24678050}}

45 {10146899, 1, {146511208}}

46 {10233389, 1, {88593477}}

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

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

49 {14788889, 1, {912985153}}

50 {20248999, 1, {534494836}}

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

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

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

Links

• Hans Havermann, Table of n, a(n) for n = 1..51
• Eric Angelini, A Recurring Digital Invariant Variant
• E. Angelini, A Recurring Digital Invariant Variant [Cached, with permission]

Crossrefs

Cf. A005188, A151544, A157714.