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The maximum exponent in the prime factorization of 2*n-1.
5

%I #8 Jul 29 2024 06:17:56

%S 0,1,1,1,2,1,1,1,1,1,1,1,2,3,1,1,1,1,1,1,1,1,2,1,2,1,1,1,1,1,1,2,1,1,

%T 1,1,1,2,1,1,4,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,2,1,2,1,3,1,1,1,1,3,

%U 1,1,1,1,1,2,1,1,2,1,1,1,1,1,1,1,2,2,1,2,1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,2,1

%N The maximum exponent in the prime factorization of 2*n-1.

%H Amiram Eldar, <a href="/A375039/b375039.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.

%F a(n) = A051903(2*n-1).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{k>=2} (1 - 1/((1-1/2^k) * zeta(k))) = 1.25979668632898014495... .

%t a[n_] := Max[FactorInteger[2*n - 1][[;; , 2]]]; a[1] = 0; Array[a, 100]

%o (PARI) a(n) = if(n == 1, 0, vecmax(factor(2*n-1)[,2]));

%Y Bisection of A051903.

%Y Cf. A033150, A375040.

%K nonn,easy

%O 1,5

%A _Amiram Eldar_, Jul 28 2024