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%I #31 Aug 28 2022 08:28:14
%S 1,1,3,8,17,41,92,215,487,1126,2583,5981,13698,31647,72678,167474,
%T 385021,887133,2041375,4700526,10817997,24908164,57334111,131995229
%N Number of terms on the n-th level of the Inverse-Totient-Tree (ITT). The 0th, 1st, 2nd and 3rd levels are {1}, {2}, {3, 4, 6}, {5, 7, 8, 9, 10, 12, 14, 18} with 1, 1, 3, 8 entries resp. The (n+1)-st level is obtained from the n-th level as the union of the inverse phi sets of the terms occurring earlier on the n-th level.
%C Level 0 is the set {1}, and level k>=1 is the set of numbers t such that phi(t) is in the set of level k; a(n) is the cardinality of the set in level n. - _Joerg Arndt_, Jan 07 2015
%C The 3rd level is {5, 8, 10, 12, 7, 9, 14, 18} and a(3)=8. Generate invphi(5)={}, .., invphi(12)={13, 21, 26, 28, 36, 42}, ..., invphi(14)={}, .. The union of these inverses gives the 4th Floor ={15, 16, 20, 24, 30, 11, 22, 13, 21, 26, 28, 36, 42, 19, 27, 38, 54}, which has 17 terms. So a(4)=17. Each level-set may have entries either from A007617, A005278 (initial nodes of the tree) or from A000010 (invphi-continuable numbers).
%C Results of Shapiro show that largest number in the n-th level is 2*3^(n-1). The Mathematica code first computes A003434(k) for k <= 2*3^(n-1); then it gives the number of numbers k for which A003434(k) = n. - _T. D. Noe_, Mar 08 2004
%H Max Alekseyev, <a href="http://home.gwu.edu/~maxal/gpscripts/">PARI scripts for various problems</a>
%H Harold Shapiro, <a href="http://www.jstor.org/stable/2303988">An arithmetic function arising from the phi function</a>, Amer. Math. Monthly, Vol. 50, No. 1 (1943), 18-30.
%F a(n) = Cardinality[Floor(n)], where Floor(0)={1}, Floor(n+1)=Union[invphi(t(i, n)), i=1..a(n))], where t(i, n) is the i-th integer in Floor(n), ordered by size or lexicographically.
%t Table[ Length[ Select[ Range[ 1, 1050000 ], Equal[ flo[ # ], k ]& ] ], {k, 1, 20} ], where flo[ x_ ] := Length[ Delete[ FixedPointList[ EulerPhi, x ], -1 ] ]
%t nMax=16; kMax=2*3^(nMax-1); a=Table[0, {kMax}]; Do[e=EulerPhi[k]; a[[k]]=1+a[[e]], {k, 2, kMax}]; Table[Count[a, _?(#==n &)], {n, 0, nMax}]
%o (PARI) lista(nn) = {my(v = [1]); print1(#v, ", "); for (n=1, nn, my(nv = []); for (i=1, #v, nv = Set(concat(nv, invphi(v[i])));); nv = setminus(nv, v); print1(#nv, ", "); v = nv;);} \\ _Michel Marcus_, Sep 02 2019
%Y Cf. A000010, A007617, A005278.
%Y Cf. A003434 (iterations of phi(n) needed to reach 1).
%K nonn,more
%O 0,3
%A _Labos Elemer_, Jan 03 2001
%E a(13)-a(16) from _T. D. Noe_, Mar 08 2004
%E a(17)-a(23) from _Sean A. Irvine_, Aug 28 2022
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