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).

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

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: A240086 A322306 A359778 * A093914 A007061 A001817

Adjacent sequences: A305827 A305828 A305829 * A305831 A305832 A305833

Keyword

nonn

Author

Gus Wiseman, Jun 10 2018

Status

approved