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%I #21 Dec 02 2023 09:50:37
%S 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,
%T 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,
%U 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
%N Length of iteration sequence if function A056239, a pseudo-logarithm is iterated and started at n. Fixed point equals zero for all initial values.
%C From _Gus Wiseman_, Dec 01 2023: (Start)
%C Conjecture:
%C - The position of first appearance of k is n = A007097(k-2).
%C - The position of last appearance of k is n = A014221(k-2) = 2^^(k-2).
%C - The number of times k appears is: 1, 1, 2, 8, 435, ...
%C (End)
%D 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.
%D 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.
%H Alois P. Heinz, <a href="/A082090/b082090.txt">Table of n, a(n) for n = 1..1000</a>
%e n=127:list={127,31,11,5,3,2,1,0},a[127]=8
%p f:= n-> add (numtheory[pi](i[1])*i[2], i=ifactors(n)[2]):
%p a:= n-> 1+ `if`(n=1, 1, a(f(n))):
%p seq (a(n), n=1..120); # _Alois P. Heinz_, Aug 09 2012
%t 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}]
%t Table[Length[FixedPointList[Total[PrimePi/@Join@@ ConstantArray@@@FactorInteger[#]]&,n]]-1, {n,100}] (* _Gus Wiseman_, Dec 01 2023 *)
%Y Cf. A008475, A001414, A082083-A082086, A082091.
%Y A112798 lists prime indices, length A001222, sum A056239.
%Y A304038 lists distinct prime indices, length A001221, sum A066328.
%Y Cf. A000079, A000720, A052410, A181819, A225485, A238747, A288636, A296150.
%K nonn
%O 1,1
%A _Labos Elemer_, Apr 09 2003
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