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

1,2

Comments

The old entry with this A-number was a duplicate of A030298.

a(52) is currently unknown. - Donovan Johnson

a(52)-a(10000) were found using a conjunction of Mathematica and Kim Walisch's primecount program. The additional values of the prime-counting function can be found in the second a-file. - Matthew House, Dec 23 2016

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

Example

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)

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 *)

Prog

(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

Crossrefs

Cf. A087712, A007097, A077960. See also A145077, A145078, A145079, A144760, A144813, A144814, A144915, A144914.

See A156055 for another version.

Sequence in context: A156055 A096357 A091507 * A034855 A105214 A358940

Adjacent sequences: A098279 A098280 A098281 * A098283 A098284 A098285

Keyword

nonn,base,nice

Author

Eric Angelini, Feb 02 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

Status

approved