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A306446
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a(n) is the number of connected components in the Fermi-Dirac factorization of n (see Comments for precise definition).
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1
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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
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OFFSET
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1,10
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COMMENTS
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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
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LINKS
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FORMULA
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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.
(1) a(1) = 0, otherwise a(n) > 0.
For any k, n > 0:
(2b) a(A059896(n,k)) <= a(n) + a(k);
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:
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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EXAMPLE
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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.
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PROG
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(PARI) See Links section.
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CROSSREFS
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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.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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