A082090 Length of iteration sequence if function A056239, a pseudo-logarithm is iterated and started at n. Fixed point equals zero for all initial values.

2, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 5, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 6, 7, 6, 6, 6, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 6, 7, 6, 6, 6, 6, 6, 7, 6, 7, 6, 7, 6, 7, 7, 6, 6, 6, 6, 7, 6, 6, 7, 7, 6, 6, 7, 6, 6, 7, 6, 6, 7, 7, 6, 7, 6, 7, 6, 6, 6, 7, 6, 7, 6, 6, 7

Offset: 1

Author

Labos Elemer, Apr 09 2003

Keywords

• nonn

Comments

From Gus Wiseman, Dec 01 2023: (Start)

Conjecture:

- The position of first appearance of k is n = A007097(k-2).

- The position of last appearance of k is n = A014221(k-2) = 2^^(k-2).

- The number of times k appears is: 1, 1, 2, 8, 435, ...

(End)

Examples

			n=127:list={127,31,11,5,3,2,1,0},a[127]=8
		

References

• Mohammad K. Azarian, On the Fixed Points of a Function and the Fixed Points of its Composite Functions, International Journal of Pure and Applied Mathematics, Vol. 46, No. 1, 2008, pp. 37-44. Mathematical Reviews, MR2433713 (2009c:65129), March 2009. Zentralblatt MATH, Zbl 1160.65015.
• Mohammad K. Azarian, Fixed Points of a Quadratic Polynomial, Problem 841, College Mathematics Journal, Vol. 38, No. 1, January 2007, p. 60. Solution published in Vol. 39, No. 1, January 2008, pp. 66-67.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A008475, A001414, A082083-A082086, A082091.

A112798 lists prime indices, length A001222, sum A056239.

A304038 lists distinct prime indices, length A001221, sum A066328.

Cf. A000079, A000720, A052410, A181819, A225485, A238747, A288636, A296150.

Programs

Maple

f:= n-> add (numtheory[pi](i[1])*i[2], i=ifactors(n)[2]):
a:= n-> 1+ `if`(n=1, 1, a(f(n))):
seq (a(n), n=1..120);  # Alois P. Heinz, Aug 09 2012

Mathematica

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}] bpi[x_] := Table[PrimePi[Part[ba[x], j]], {j, 1, lf[x]}] api[x_] := Apply[Plus, ep[x]*bpi[x]] Table[Length[FixedPointList[api, w]]-1, {w, 2, 128}]
Table[Length[FixedPointList[Total[PrimePi/@Join@@ ConstantArray@@@FactorInteger[#]]&,n]]-1, {n,100}] (* Gus Wiseman, Dec 01 2023 *)