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 A279614 a(1)=1, a(d(x_1)*..*d(x_k)) = 1+a(x_1)+..+a(x_k) where d(n) = n-th Fermi-Dirac prime. 8
 1, 2, 3, 4, 5, 4, 6, 5, 5, 6, 7, 6, 6, 7, 7, 6, 7, 6, 8, 8, 8, 8, 7, 7, 7, 7, 7, 9, 8, 8, 8, 7, 9, 8, 10, 8, 7, 9, 8, 9, 8, 9, 7, 10, 9, 8, 9, 8, 9, 8, 9, 9, 9, 8, 11, 10, 10, 9, 9, 10, 8, 9, 10, 9, 10, 10, 8, 10, 9, 11, 8, 9, 8, 8, 9, 11, 12, 9, 8, 10, 10, 9 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS A Fermi-Dirac prime (A050376) is a positive integer of the form p^(2^k) where p is prime and k>=0. In analogy with the Matula-Goebel correspondence between rooted trees and positive integers (see A061775), the iterated normalized Fermi-Dirac representation gives a correspondence between rooted identity trees and positive integers. Then a(n) is the number of nodes in the rooted identity tree corresponding to n. LINKS OEIS Wiki, "Fermi-Dirac representation" of n FORMULA Number of appearances of n is |a^{-1}(n)| = A004111(n). EXAMPLE Sequence of rooted identity trees represented as finitary sets begins: {}, {{}}, {{{}}}, {{{{}}}}, {{{{{}}}}}, {{}{{}}}, {{{{{{}}}}}}, {{}{{{}}}}, {{{}{{}}}}, {{}{{{{}}}}}, {{{{{{{}}}}}}}, {{{}}{{{}}}}, {{{}{{{}}}}}, {{}{{{{{}}}}}}, {{{}}{{{{}}}}}, {{{{}{{}}}}}, {{{}{{{{}}}}}}, {{}{{}{{}}}}, {{{{{{{{}}}}}}}}, {{{{}}}{{{{}}}}}, {{{}}{{{{{}}}}}}, {{}{{{{{{}}}}}}}, {{{{}}{{{}}}}}, {{}{{}}{{{}}}}. MATHEMATICA nn=200; FDfactor[n_]:=If[n===1, {}, Sort[Join@@Cases[FactorInteger[n], {p_, k_}:>Power[p, Cases[Position[IntegerDigits[k, 2]//Reverse, 1], {m_}->2^(m-1)]]]]]; FDprimeList=Array[FDfactor, nn, 1, Union]; FDweight[n_?(#<=nn&)]:=If[n===1, 1, 1+Total[FDweight[Position[FDprimeList, #][[1, 1]]]&/@FDfactor[n]]]; Array[FDweight, nn] CROSSREFS Cf. A004111, A050376, A061773, A061775, A084400, A276625, A279065. Sequence in context: A238288 A323161 A152739 * A212639 A212647 A303233 Adjacent sequences:  A279611 A279612 A279613 * A279615 A279616 A279617 KEYWORD nonn AUTHOR Gus Wiseman, Dec 15 2016 STATUS approved

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Last modified October 22 00:52 EDT 2019. Contains 328315 sequences. (Running on oeis4.)