cp's OEIS Frontend

A375670 The maximum exponent in the prime factorization of the largest 5-rough divisor of n.

%I A375670 #13 Sep 13 2024 10:14:26
%S A375670 0,0,0,0,1,0,1,0,0,1,1,0,1,1,1,0,1,0,1,1,1,1,1,0,2,1,0,1,1,1,1,0,1,1,
%T A375670 1,0,1,1,1,1,1,1,1,1,1,1,1,0,2,2,1,1,1,0,1,1,1,1,1,1,1,1,1,0,1,1,1,1,
%U A375670 1,1,1,0,1,1,2,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,2,1,2,1,1,1,1,1
%N A375670 The maximum exponent in the prime factorization of the largest 5-rough divisor of n.
%C A375670 First differs from A106799 at n = 35.
%C A375670 The largest exponent among the exponents of the primes that are larger than 3 in the prime factorization of n.
%H A375670 Amiram Eldar, <a href="/A375670/b375670.txt">Table of n, a(n) for n = 1..10000</a>
%H A375670 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F A375670 a(n) = A051903(A065330(n)).
%F A375670 a(n) = 0 if and only if n is a 3-smooth number (A003586).
%F A375670 a(n) = 1 if and only if n is a product of a squarefree 5-rough number larger than 1 and a 3-smooth number.
%F A375670 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} k * d(k) = 1.1034178389191320571029... , where d(k) is the asymptotic density of the occurrences of k in this sequence: d(1) = 3/(2*zeta(2)), and d(k) = (1/zeta(k+1)) / ((1-1/2^(k+1))*(1-1/3^(k+1))) - (1/zeta(k)) / ((1-1/2^k)*(1-1/3^k)) for k >= 2.
%F A375670 In general, the asymptotic mean of the maximum exponent in the prime factorization of the largest p-rough divisor of n is Sum_{k>=1} k * d(k), where d(1) = 1/(zeta(2) * f(p, 2)), d(k) = 1/(zeta(k+1) * f(p, k+1)) - 1/(zeta(k) * f(p, k)) for k >= 2, and f(p, m) = Product_{q prime < p} (1-1/q^m).
%t A375670 a[n_] := Module[{m = n / Times@@({2,3}^IntegerExponent[n,{2,3}])}, If[m == 1, 0, Max[FactorInteger[m][[;; , 2]]]]]; Array[a, 100]
%o A375670 (PARI) a(n) = {my(m = n >> valuation(n, 2)/3^valuation(n, 3)); if(m == 1, 0,vecmax(factor(m)[,2]));}
%Y A375670 Cf. A003586, A007310 (5-rough numbers), A051903, A065330, A106799, A375669.
%K A375670 nonn,easy
%O A375670 1,25
%A A375670 _Amiram Eldar_, Aug 23 2024