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

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, 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, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 4, 16, 1, 1, 2, 1, 1, 1, 3, 1, 2

Offset: 1

Author

Ilya Gutkovskiy, Mar 19 2025

Keywords

• nonn

Comments

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A005361, A033150, A051903, A051904, A052485, A062977, A066048, A304233, A333352.

Programs

Mathematica

Table[Max @@ (#[[2]] & /@ FactorInteger[n]) Min @@ (#[[2]] & /@ FactorInteger[n]), {n, 90}]

PARI

a(n) = if(n == 1, 1, my(e = factor(n)[,2]); vecmin(e) * vecmax(e)); \\ Amiram Eldar, Mar 28 2025

Formula

If n = Product (p_j^k_j) then a(n) = min{k_j} * max{k_j}.

a(n) = A051903(n) * A051904(n) for n > 1.