login
Largest exponents of perfect powers (m^k where m is an integer and k >= 2) multiplied by -1 when base m is prime (m^k thus a prime power).
3

%I #12 Oct 05 2024 22:15:25

%S 2,-2,-3,-2,-4,-2,-3,-5,2,-2,-6,-4,2,-2,-3,-7,2,-2,2,3,2,-5,-8,-2,2,

%T -3,-2,2,2,2,-9,-2,2,-4,2,-6,2,-2,2,-2,3,-10,2,2,2,4,-3,-2,2,2,2,-2,3,

%U 2,-2,2,2,-11,2,-7,-3,-2,2,-4,2,2,2,3,-2,2,2,-5,2,2,2,3,-2,2,-2,2,2,-12,2,2

%N Largest exponents of perfect powers (m^k where m is an integer and k >= 2) multiplied by -1 when base m is prime (m^k thus a prime power).

%H Daniel Forgues, <a href="/A157986/b157986.txt">Table of n, a(n) for n=1..10000</a>

%F a(n) = {k}_n * (-1)^(Pi(m) - Pi(m-1)) where {k}_n is the exponent of {m^k}_n (the n-th perfect power with positive integer base m corresponding to largest integer exponent k) and Pi(m) is the prime counting function evaluated at m.

%F a(n) = A025479(n) * (-1)^{Pi(m) - Pi(m-1)}, with m = A001597(n)^(1/(A025479(n))).

%Y Cf. A001597 (perfect powers), A025479 (largest exponents of perfect powers).

%Y Cf. A025478 (least roots of perfect powers).

%Y Cf. A157985.

%K sign

%O 1,1

%A _Daniel Forgues_, Mar 10 2009