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

1,4

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.

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.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A387716 A382889 A084885 * A360969 A112538 A008477

Adjacent sequences: A382216 A382217 A382218 * A382220 A382221 A382222

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 19 2025

Status

approved