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A306446 a(n) is the number of connected components in the Fermi-Dirac factorization of n (see Comments for precise definition).

%I A306446 #29 Jan 24 2021 13:30:29
%S A306446 0,1,1,1,1,1,1,1,1,2,1,2,1,2,1,1,1,2,1,2,2,2,1,1,1,2,1,2,1,1,1,2,2,2,
%T A306446 1,1,1,2,2,2,1,2,1,2,2,2,1,2,1,2,2,2,1,1,2,2,2,2,1,2,1,2,2,1,2,2,1,2,
%U A306446 2,2,1,1,1,2,2,2,1,2,1,2,1,2,1,3,2,2,2
%N A306446 a(n) is the number of connected components in the Fermi-Dirac factorization of n (see Comments for precise definition).
%C A306446 For any n > 0:
%C A306446 - let F(n) be the set of distinct Fermi-Dirac primes (A050376) with product n,
%C A306446 - let G(n) be the undirected graph with vertices F(n) and the following connection rules:  for any k >= 0 and any pair of consecutive prime numbers (p, q):
%C A306446    - p^(2^k) and p^(2^(k+1)) are connected,
%C A306446    - p^(2^k) and q^(2^k) are connected,
%C A306446 - a(n) is the number of connected components in G(n).
%C A306446 The sequence may be specified algebraically by formulas (1) to (2c) in my contemporary entry in the formula section. - _Peter Munn_, Jan 05 2021
%H A306446 Antti Karttunen, <a href="/A306446/b306446.txt">Table of n, a(n) for n = 1..100000</a>
%H A306446 OEIS Wiki, <a href="/wiki/%22Fermi-Dirac_representation%22_of_n">"Fermi-Dirac representation" of n</a>
%H A306446 Rémy Sigrist, <a href="/A306446/a306446.gp.txt">PARI program for A306446</a>
%H A306446 Wikipedia, <a href="https://en.wikipedia.org/wiki/Connected_component_(graph_theory)">Connected component (graph theory)</a>
%F A306446 If m and n are coprime, then a(m * n) <= a(m) + a(n).
%F A306446 a(p^k) = A069010(k) for any k >= 0 and any prime number p.
%F A306446 a(n) <= A064547(n).
%F A306446 a(A002110(k)) = 1 for any k > 0.
%F A306446 a(A066205(k)) = k for any k > 0.
%F A306446 From _Peter Munn_, Jan 05 2021: (Start)
%F A306446 (1) a(1) = 0, otherwise a(n) > 0.
%F A306446 For any k, n > 0:
%F A306446 (2a) a(A050376(k)) = 1;
%F A306446 (2b) a(A059896(n,k)) <= a(n) + a(k);
%F A306446 (2c) a(A059896(n,k)) = a(n) + a(k) if and only if A059895(A306697(n,24), k) = 1 and A059895(n, A306697(k,24)) = 1.
%F A306446 For any n > 0, write n = j * k^2 * m^4, j, k squarefree, m > 0:
%F A306446 (3a) a(n) <= a(j) + a(k) + a(m);
%F A306446 (3b) if gcd(j, k) = 1, a(n) = a(j) + a(n/j);
%F A306446 (3c) if gcd(j, k) = j, a(n) = a(n/j);
%F A306446 (3d) if gcd(k, m) = 1, a(n) = a(n/m^4) + a(m^4);
%F A306446 (3e) if gcd(j, k) = k and gcd(k, m) = 1, a(n) = a(j) + a(m).
%F A306446 For any n > 0:
%F A306446 (4a) a(n^2) = a(A003961(n)) = a(A225546(n)) = a(n);
%F A306446 (4b) a(n) = a(A340346(n)) + a(n/A340346(n)).
%F A306446 For any odd n > 0 (with k >= 0, m >= 0):
%F A306446 (5) If n = 9^k * (6m + 1) or n = 9^k * (6m + 5) then a(2n) = a(n) + 1; otherwise a(2n) = a(n).
%F A306446 (End)
%e A306446 For n = 67!:
%e A306446 - the Fermi-Dirac primes p^(2^k) in F(67!) can be depicted as:
%e A306446     6|@
%e A306446     5|
%e A306446     4| @
%e A306446     3| @@@
%e A306446     2| @@ @@
%e A306446     1| @@@@ @@@@@
%e A306446     0| @@  @@@   @@@@@@@@
%e A306446   ---+-------------------
%e A306446   k/p|    111122334445566
%e A306446      |2357137939171373917
%e A306446 - G(67!) has 4 connected components:
%e A306446     6|A
%e A306446     5|
%e A306446     4| B
%e A306446     3| BBB
%e A306446     2| BB BB
%e A306446     1| BBBB CCCCC
%e A306446     0| BB  CCC   DDDDDDDD
%e A306446   ---+-------------------
%e A306446   k/p|    111122334445566
%e A306446      |2357137939171373917
%e A306446 - hence a(67!) = 4.
%o A306446 (PARI) See Links section.
%Y A306446 Cf. A064547, A069010.
%Y A306446 A050376, A059895, A059896, A306697 are used in a formula defining this sequence.
%Y A306446 A329050 corresponds to the array depicted in the first example, with prime(n+1) = p.
%Y A306446 The formula section details how the sequence maps the terms of A002110, A066205.
%Y A306446 A003961, A225546, A340346 are used to express relationship between terms of this sequence.
%K A306446 nonn
%O A306446 1,10
%A A306446 _Rémy Sigrist_, Feb 16 2019