A306697 -id:A306697 - cp's OEIS frontend

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

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, 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, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 2, 2, 2

Offset: 1

Rémy Sigrist, Feb 16 2019

nonn

For any n > 0:
- let F(n) be the set of distinct Fermi-Dirac primes (A050376) with product n,
- 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):
   - p^(2^k) and p^(2^(k+1)) are connected,
   - p^(2^k) and q^(2^k) are connected,
- a(n) is the number of connected components in G(n).
The sequence may be specified algebraically by formulas (1) to (2c) in my contemporary entry in the formula section. - Peter Munn, Jan 05 2021

			For n = 67!:
- the Fermi-Dirac primes p^(2^k) in F(67!) can be depicted as:
    6|@
    5|
    4| @
    3| @@@
    2| @@ @@
    1| @@@@ @@@@@
    0| @@  @@@   @@@@@@@@
  ---+-------------------
  k/p|    111122334445566
     |2357137939171373917
- G(67!) has 4 connected components:
    6|A
    5|
    4| B
    3| BBB
    2| BB BB
    1| BBBB CCCCC
    0| BB  CCC   DDDDDDDD
  ---+-------------------
  k/p|    111122334445566
     |2357137939171373917
- hence a(67!) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..100000
OEIS Wiki, "Fermi-Dirac representation" of n
Rémy Sigrist, PARI program for A306446
Wikipedia, Connected component (graph theory)

Cf. A064547, A069010.
A050376, A059895, A059896, A306697 are used in a formula defining this sequence.
A329050 corresponds to the array depicted in the first example, with prime(n+1) = p.
The formula section details how the sequence maps the terms of A002110, A066205.
A003961, A225546, A340346 are used to express relationship between terms of this sequence.

PARI
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See Links section.
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If m and n are coprime, then a(m * n) <= a(m) + a(n).
a(p^k) = A069010(k) for any k >= 0 and any prime number p.
a(n) <= A064547(n).
a(A002110(k)) = 1 for any k > 0.
a(A066205(k)) = k for any k > 0.
From Peter Munn, Jan 05 2021: (Start)
(1) a(1) = 0, otherwise a(n) > 0.
For any k, n > 0:
(2a) a(A050376(k)) = 1;
(2b) a(A059896(n,k)) <= a(n) + a(k);
(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.
For any n > 0, write n = j * k^2 * m^4, j, k squarefree, m > 0:
(3a) a(n) <= a(j) + a(k) + a(m);
(3b) if gcd(j, k) = 1, a(n) = a(j) + a(n/j);
(3c) if gcd(j, k) = j, a(n) = a(n/j);
(3d) if gcd(k, m) = 1, a(n) = a(n/m^4) + a(m^4);
(3e) if gcd(j, k) = k and gcd(k, m) = 1, a(n) = a(j) + a(m).
For any n > 0:
(4a) a(n^2) = a(A003961(n)) = a(A225546(n)) = a(n);
(4b) a(n) = a(A340346(n)) + a(n/A340346(n)).
For any odd n > 0 (with k >= 0, m >= 0):
(5) If n = 9^k * (6m + 1) or n = 9^k * (6m + 5) then a(2n) = a(n) + 1; otherwise a(2n) = a(n).
(End)

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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).

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