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

1,1

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)

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

Example

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

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 *)

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.

Sequence in context: A372599 A030602 A133947 * A060197 A116487 A249041

Adjacent sequences: A082087 A082088 A082089 * A082091 A082092 A082093

Keyword

nonn

Author

Labos Elemer, Apr 09 2003

Status

approved