A157986 -id:A157986 - cp's OEIS frontend

A157985 Perfect powers (m^k where m is an integer and k >= 2) multiplied by -1 when m is prime for largest k (m^k thus a prime power).

1, -4, -8, -9, -16, -25, -27, -32, 36, -49, -64, -81, 100, -121, -125, -128, 144, -169, 196, 216, 225, -243, -256, -289, 324, -343, -361, 400, 441, 484, -512, -529, 576, -625, 676, -729, 784, -841, 900, -961, 1000, -1024, 1089, 1156, 1225, 1296, -1331

Offset: 1

Daniel Forgues, Mar 10 2009

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The rather strange phrase "largest k" in the definition refers to the fact that there can be several ways to write a number in the form m^k. - N. J. A. Sloane, Jan 01 2019

Daniel Forgues, Table of n, a(n) for n=1..10000

Cf. A001597 (perfect powers), A025479 (largest exponents of perfect powers).
Cf. A025478 (least roots of perfect powers).
Cf. A157986.

a(n) = {m^k}_n * (-1)^(Pi(m) - Pi(m-1)) where {m^k}_n is 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.

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

A157987 Smallest roots m of perfect powers (m^k where m is an integer and k >= 2) multiplied by -1 when m is prime (m^k thus a prime power).

Original entry on oeis.org

1, -2, -2, -3, -2, -5, -3, -2, 6, -7, -2, -3, 10, -11, -5, -2, 12, -13, 14, 6, 15, -3, -2, -17, 18, -7, -19, 20, 21, 22, -2, -23, 24, -5, 26, -3, 28, -29, 30, -31, 10, -2, 33, 34, 35, 6, -11, -37, 38, 39, 40, -41, 12, 42, -43, 44, 45, -2, 46, -3, -13, -47, 48, -7, 50, 51, 52

Offset: 1

Daniel Forgues, Mar 10 2009, Mar 14 2009

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Daniel Forgues, Table of n, a(n) for n=1..10000

Cf. A157985 Perfect powers (m^k where m is an integer and k >= 2) multiplied by -1 when m is prime for largest k (m^k thus a prime power).
Cf. A157986 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).
Cf. A001597 Perfect powers: m^k where m is an integer and k >= 2.
Cf. A025479 Largest exponents of perfect powers (A001597).
Cf. A025478 Least roots of perfect powers (A001597).

a(n) = {m}_n * (-1)^{Pi(m) - Pi(m-1)}
where {m}_n is the smallest root 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.
a(n) = m * (-1)^{Pi(m) - Pi(m-1)}, with m = A025478(n) = {A001597(n)}^{1/{A025479(n)}}.

cp's OEIS Frontend

A157985 Perfect powers (m^k where m is an integer and k >= 2) multiplied by -1 when m is prime for largest k (m^k thus a prime power).

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