login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Table of n, a(n) for n=1..82.

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: A244904 A238288 A152739 * A212639 A212647 A303233

Adjacent sequences:  A279611 A279612 A279613 * A279615 A279616 A279617

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 15 2016

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 21 19:08 EST 2019. Contains 319350 sequences. (Running on oeis4.)