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A305830 Combined weight of the n-th FDH set-system. Factor n into distinct Fermi-Dirac primes (A050376), normalize by replacing every instance of the k-th Fermi-Dirac prime with k, then add up their FD-weights (A064547). 5
0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 3, 2, 2, 2, 2, 1, 2, 2, 2, 3, 1, 1, 3, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 1, 3, 3, 2, 3, 2, 2, 2, 2, 2, 3, 1, 2, 3, 2, 3, 2, 3, 3, 3, 2, 1, 3, 2, 1, 2, 2, 2, 3, 2, 2, 2, 1, 1, 3, 3, 2, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,9

COMMENTS

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. Every positive integer n has a unique factorization of the form n = f(s_1)*...*f(s_k) where the s_i are strictly increasing positive integers. Then a(n) = w(s_1) + ... + w(s_k) where w = A064547.

LINKS

Table of n, a(n) for n=1..87.

EXAMPLE

Sequence of FDH set-systems (a list containing all finite sets of finite sets of positive integers) begins:

   1: {}

   2: {{}}

   3: {{1}}

   4: {{2}}

   5: {{3}}

   6: {{},{1}}

   7: {{4}}

   8: {{},{2}}

   9: {{1,2}}

  10: {{},{3}}

  11: {{5}}

  12: {{1},{2}}

  13: {{1,3}}

  14: {{},{4}}

  15: {{1},{3}}

  16: {{6}}

  17: {{1,4}}

  18: {{},{1,2}}

  19: {{7}}

  20: {{2},{3}}

  21: {{1},{4}}

  22: {{},{5}}

  23: {{2,3}}

  24: {{},{1},{2}}

  25: {{8}}

  26: {{},{1,3}}

  27: {{1},{1,2}}

MATHEMATICA

nn=100;

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]; FDrules=MapIndexed[(#1->#2[[1]])&, FDprimeList];

Table[Total[Length/@(FDfactor/@(FDfactor[n]/.FDrules))], {n, nn}]

CROSSREFS

Cf. A003963, A050376, A056239, A061775, A064547, A213925, A279065, A279614, A299755, A299756, A299757, A302242, A302243, A305829, A305832.

Sequence in context: A206719 A240086 A322306 * A093914 A007061 A001817

Adjacent sequences:  A305827 A305828 A305829 * A305831 A305832 A305833

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 10 2018

STATUS

approved

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Last modified November 18 17:45 EST 2019. Contains 329287 sequences. (Running on oeis4.)