cp's OEIS Frontend

A382219 Product of the largest and smallest exponents in the prime factorization of n.

%I A382219 #12 Mar 28 2025 08:00:20
%S A382219 1,1,1,4,1,1,1,9,4,1,1,2,1,1,1,16,1,2,1,2,1,1,1,3,4,1,9,2,1,1,1,25,1,
%T A382219 1,1,4,1,1,1,3,1,1,1,2,2,1,1,4,4,2,1,2,1,3,1,3,1,1,1,2,1,1,2,36,1,1,1,
%U A382219 2,1,1,1,6,1,1,2,2,1,1,1,4,16,1,1,2,1,1,1,3,1,2
%N A382219 Product of the largest and smallest exponents in the prime factorization of n.
%C A382219 The asymptotic density of the occurrences of k = 1, 2, ... in this sequence is 1/zeta(2) for k = 1 and 1/zeta(k+1) - 1/zeta(k) for k >= 2, and the asymptotic mean of this sequence is A033150, the same densities and mean as in A051903, since a(n) = A051903(n) for nonpowerful numbers n (A052485) whose asymptotic density is 1. - _Amiram Eldar_, Mar 28 2025
%H A382219 Amiram Eldar, <a href="/A382219/b382219.txt">Table of n, a(n) for n = 1..10000</a>
%H A382219 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F A382219 If n = Product (p_j^k_j) then a(n) = min{k_j} * max{k_j}.
%F A382219 a(n) = A051903(n) * A051904(n) for n > 1.
%t A382219 Table[Max @@ (#[[2]] & /@ FactorInteger[n]) Min @@ (#[[2]] & /@ FactorInteger[n]), {n, 90}]
%o A382219 (PARI) a(n) = if(n == 1, 1, my(e = factor(n)[,2]); vecmin(e) * vecmax(e)); \\ _Amiram Eldar_, Mar 28 2025
%Y A382219 Cf. A005361, A033150, A051903, A051904, A052485, A062977, A066048, A304233, A333352.
%K A382219 nonn
%O A382219 1,4
%A A382219 _Ilya Gutkovskiy_, Mar 19 2025