A098282 Iterate the map k -> A087712(k) starting at n; a(n) is the number of steps at which we see a repeated term for the first time; or -1 if the trajectory never repeats.
1, 2, 3, 6, 4, 31, 7, 55, 4, 33, 5, 30, 32, 1, 4, 19, 8, 112, 56, 16, 27, 4, 4, 26, 2, 20, 223, 102, 34, 14, 6, 162, 2, 9, 10, 75, 31, 113, 21, 100, 33, 20, 2, 23, 30, 57, 5, 28, 24, 30, 224, 269, 20, 295, 11, 85, 103, 140, 9, 71, 113, 55, 34, 110, 76, 49, 57
Offset: 1
Examples
1 -> 1; 1 step to see a repeat, so a(1) = 1. 2 -> 1 -> 1; 2 steps to see a repeat. 3 -> 2 -> 1 -> 1; 3 steps to see a repeat. 4 -> 11 -> 5 -> 3 -> 2 -> 1 -> 1; 6 steps to see a repeat. 6 -> 12 -> 112 -> 11114 -> 1733 -> 270 -> 12223 -> 7128 -> 11122225 -> 33991010 -> 13913661 -> 2107998 -> 12222775 -> 33910130 -> 131212367 -> 56113213 -> 6837229 -> 4201627 -> 266366 -> 112430 -> 131359 -> 7981 -> 969 -> 278 -> 134 -> 119 -> 47 -> 15 -> 23 -> 9 -> 22 -> 15; 31 steps to see a repeat. 9 -> 22 -> 15 -> 23 -> 9; 4 steps to see a repeat. From _David Applegate_ and _N. J. A. Sloane_, Feb 09 2009: (Start) The trajectories of the numbers 1 through 17, up to and including the first repeat, are as follows. Note that a(n) is one less than the number of terms shown. [1, 1] [2, 1, 1] [3, 2, 1, 1] [4, 11, 5, 3, 2, 1, 1] [5, 3, 2, 1, 1] [6, 12, 112, 11114, 1733, 270, 12223, 7128, 11122225, 33991010, 13913661, 2107998, 12222775, 33910130, 131212367, 56113213, 6837229, 4201627, 266366, 112430, 131359, 7981, 969, 278, 134, 119, 47, 15, 23, 9, 22, 15] [7, 4, 11, 5, 3, 2, 1, 1] [8, 111, 212, 1116, 112211, 52626, 124441, 28192, 11111152, 111165448, 1117261018, 1910112963, 252163429, 42205629, 2914219, 454002, 127605, 231542, 110938, 15631, 44510, 13605, 23155, 3582, 12246, 12637, 1509, 296, 11112, 111290, 131172, 1127117, 76613, 9470, 13161, 21328, 11111114, 14142115, 3625334, 1125035, 348169, 78151, 11369, 1373, 220, 1135, 349, 70, 134, 119, 47, 15, 23, 9, 22, 15] [9, 22, 15, 23, 9] [10, 13, 6, 12, 112, 11114, 1733, 270, 12223, 7128, 11122225, 33991010, 13913661, 2107998, 12222775, 33910130, 131212367, 56113213, 6837229, 4201627, 266366, 112430, 131359, 7981, 969, 278, 134, 119, 47, 15, 23, 9, 22, 15] [11, 5, 3, 2, 1, 1] [12, 112, 11114, 1733, 270, 12223, 7128, 11122225, 33991010, 13913661, 2107998, 12222775, 33910130, 131212367, 56113213, 6837229, 4201627, 266366, 112430, 131359, 7981, 969, 278, 134, 119, 47, 15, 23, 9, 22, 15] [13, 6, 12, 112, 11114, 1733, 270, 12223, 7128, 11122225, 33991010, 13913661, 2107998, 12222775, 33910130, 131212367, 56113213, 6837229, 4201627, 266366, 112430, 131359, 7981, 969, 278, 134, 119, 47, 15, 23, 9, 22, 15] [14, 14] [15, 23, 9, 22, 15] [16, 1111, 526, 156, 1126, 1103, 185, 312, 11126, 1734, 1277, 206, 127, 31, 11, 5, 3, 2, 1, 1] [17, 7, 4, 11, 5, 3, 2, 1, 1] For n = 18 see A077960. (End)
Links
- Matthew House, Table of n, a(n) for n = 1..10000
- Farideh Firoozbakht, Notes on the missing terms in this sequence
- Matthew House, Values found using primecount
Crossrefs
Programs
-
GBnums
void ea (n) { mpz u[] ; // factors mpz tr[]; // sequence print(n); while(n > 1) { lfactors(u,n); // factorize into u vmap(u,pi); // replace factors by rank n = catv(u); // concatenate print(n); if(vsearch(tr,n) > 0) break; // loop found vpush(tr,n); // remember n } println(''); } // Jacques Tramu -
Haskell
import Data.List (genericIndex) a098282 n = f [n] where f xs = if y `elem` xs then length xs else f (y:xs) where y = genericIndex (map a087712 [1..]) (head xs - 1) -- Reinhard Zumkeller, Jul 14 2013 -
Maple
with(numtheory): f := proc(n) local t1, v, r, x, j; if (n = 1) then return 1; end if; t1 := ifactors(n): v := 0; for x in op(2,t1) do r := pi(x[1]): for j from 1 to x[2] do v := v * 10^length(r) + r; end do; end do; v; end proc; t := proc(n) local v, l, s; v := n; s := {v}; l := [v]; v := f(v); while not v in s do s := s union {v}; l := [op(l),v]; v := f(v); end do; [op(l),v]; end proc; [seq(nops(t(n))-1, n=1..17)]; # David Applegate and N. J. A. Sloane, Feb 09 2009 -
Mathematica
f[n_] := If[n==1,1,FromDigits@ Flatten[ IntegerDigits@# & /@ (PrimePi@# & /@ Flatten[ Table[ First@#, {Last@#}] & /@ FactorInteger@n])]]; g[n_] := Length@ NestWhileList[f, n, UnsameQ, All] - 1; Array[g, 39] (* Robert G. Wilson v, Feb 02 2009; modified slightly by Farideh Firoozbakht, Feb 10 2009 *)
Extensions
a(8) and a(10) found by Jacques Tramu
Extended through a(39) by Robert G. Wilson v, Feb 02 2009
Terms through a(39) corrected by Farideh Firoozbakht, Feb 10 2009
a(40)-a(51) from Donovan Johnson, Jan 08 2011
More terms from and a(40) corrected by Matthew House, Dec 23 2016
Comments