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

1,4

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.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Wikipedia, Fermi-Dirac prime.

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... .

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Cf. A050377, A335428, A344417, A347437.

Sequence in context: A159631 A368216 A335428 * A050377 A344417 A347437

Adjacent sequences: A368778 A368779 A368780 * A368782 A368783 A368784

Keyword

nonn,easy

Author

Amiram Eldar, Jan 05 2024

Status

approved