1,2

a(40) is currently unknown.

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

Farideh Firoozbakht, Notes on the missing terms in this sequence

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.

Comment from David Applegate and N. J. A. Sloane (njas(AT)research.att.com), 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 program from David Applegate and N. J. A. Sloane (njas(AT)research.att.com), Feb 09 2009) (Start) 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)]; (End)

(Mma program from Robert G. Wilson, v (rgwv(AT)rgwv.com), Feb 02 2009; modified slightly by Farideh Firoozbakht (mymontain(AT)yahoo.com), Feb 10 2009)

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]

(GBnums code from Jacques Tramu (jacques.tramu(AT)echolalie.com)):

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('');

}

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 this_sequence A034855 A105214 A136315

Adjacent sequences: A098279 A098280 A098281 this_sequence A098283 A098284 A098285

nonn,base,more,nice

Eric Angelini (Eric.Angelini(AT)kntv.be), Feb 02 2009

Jacques Tramu found a(8) and a(10).

Extended through a(39) by Robert G. Wilson, v (rgwv(AT)rgwv.com), Feb 02 2009. Terms through a(39) corrected by Farideh Firoozbakht (mymontain(AT)yahoo.com), Feb 10 2009