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

%I A368781 #24 Apr 11 2025 09:50:17
%S A368781 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,
%T A368781 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,
%U A368781 1,1,1,2,1,1,2,2,1,1,1,4,4,1,1,2,1,1,1
%N A368781 The maximal exponent in the unique factorization of n in terms of distinct "Fermi-Dirac primes".
%C A368781 First differs from A335428 at n = 36. Differs from A050377, A344417 and A347437 at n = 1 and then at n = 36.
%C A368781 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.
%H A368781 Amiram Eldar, <a href="/A368781/b368781.txt">Table of n, a(n) for n = 1..10000</a>
%H A368781 Wikipedia, <a href="https://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_prime">Fermi-Dirac prime</a>.
%F A368781 a(n) = A053644(A051903(n)).
%F A368781 a(n) = 2^(A299090(n)-1) for n >= 2.
%F A368781 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... .
%F A368781 a(n) = A051903(A353897(n)). - _Amiram Eldar_, May 07 2024
%e A368781 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.
%t A368781 a[n_] := 2^Floor[Log2[Max[FactorInteger[n][[;; , 2]]]]]; a[1] = 0; Array[a, 100]
%o A368781 (PARI) a(n) = if(n > 1, 2^exponent(vecmax(factor(n)[, 2])), 0);
%o A368781 (Python)
%o A368781 from sympy import factorint
%o A368781 def A368781(n): return 1<<max(factorint(n).values()).bit_length()-1 if n>1 else 0 # _Chai Wah Wu_, Apr 11 2025
%Y A368781 Cf. A000079, A050376, A051903, A053644, A182979, A246655, A299090, A353897.
%Y A368781 Cf. A050377, A335428, A344417, A347437.
%K A368781 nonn,easy
%O A368781 1,4
%A A368781 _Amiram Eldar_, Jan 05 2024