A368781 The maximal exponent in the unique factorization of n in terms of distinct "Fermi-Dirac primes".

0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Jan 05 2024

Keywords

• nonn
• easy

Comments

First differs from A335428 at n = 36. Differs from A050377, A344417 and A347437 at n = 1 and then at n = 36.

In the unique factorization of n in terms of distinct "Fermi-Dirac primes", n is represented as a product of prime powers (A246655) whose exponents are powers of 2 (A000079). a(n) is the maximal exponent of these prime powers (and not the maximal exponent of the exponents that are powers of 2). Thus, a(n) is a power of 2 for n >= 2.

Examples

			For n = 972 = 2^2 * 3^5, the unique factorization of 972 in terms of distinct "Fermi-Dirac primes" is 2^(2^1) * 3^(2^0) * 3^(2^2). Therefore, a(972) = 2^2 = 4.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Wikipedia, Fermi-Dirac prime.

Crossrefs

Cf. A000079, A050376, A051903, A053644, A182979, A246655, A299090, A353897.

Cf. A050377, A335428, A344417, A347437.

Programs

Mathematica

a[n_] := 2^Floor[Log2[Max[FactorInteger[n][[;; , 2]]]]]; a[1] = 0; Array[a, 100]

PARI

a(n) = if(n > 1, 2^exponent(vecmax(factor(n)[, 2])), 0);

Python

from sympy import factorint
def A368781(n): return 1<1 else 0 # Chai Wah Wu, Apr 11 2025

Formula

a(n) = A053644(A051903(n)).

a(n) = 2^(A299090(n)-1) for n >= 2.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{k>=1} (2^(k-1) * (1 - 1/zeta(2^k))) = 1.56056154773294953123... .

a(n) = A051903(A353897(n)). - Amiram Eldar, May 07 2024