A158052 Largest integer power m (with 0 to stand for infinity) for which a representation of the form n = k^m exists (for some k >= 1) multiplied by -1 when k is prime.

0, -1, -1, -2, -1, 1, -1, -3, -2, 1, -1, 1, -1, 1, 1, -4, -1, 1, -1, 1, 1, 1, -1, 1, -2, 1, -3, 1, -1, 1, -1, -5, 1, 1, 1, 2, -1, 1, 1, 1, -1, 1, -1, 1, 1, 1, -1, 1, -2, 1, 1, 1, -1, 1, 1, 1, 1, 1, -1, 1, -1, 1, 1, -6, 1, 1, -1, 1, 1, 1, -1, 1, -1, 1, 1, 1, 1, 1, -1, 1, -4, 1, -1, 1, 1, 1, 1, 1

Offset

1,4

Comments

This sequence reveals, among the positive integers, which are the unit, the primes, the perfect powers (with |a(n)| as largest exponent) telling whether these are perfect powers of either primes or composites and finally which are non-perfect powers composites per the following:

a(n) < -1: perfect powers of primes (largest exponent = |a(n)|)

a(n) = -1: primes (not perfect powers)

a(n) = 0: (standing for infinity): unit, perfect power of unit

a(n) = +1: composites (not perfect powers)

a(n) > +1: perfect powers of composites (largest exponent = |a(n)|).

Links

Daniel Forgues, Table of n, a(n) for n=1..100000

Formula

a(n) = m * (-1)^{pi(k) - pi(k-1)} where m is the largest exponent of k^m = n for some k >= 1 and pi(k) is the prime counting function evaluated at k.

a(n) = A052409(n) * (-1)^{Pi(k(n)) - Pi(k(n)-1)}, with k(n) = A052410(n).

Crossrefs

Cf. A052409 a(n) = largest integer power m for which a representation of the form n = k^m exists (for some k).

Cf. A052410 Value of a in a^p=n, where p is the largest power given by A052409.

Cf. A000040 The prime numbers.

Cf. A000961 Prime powers p^k (p prime, k >= 0).

Cf. A001597 Perfect powers: m^k where m is an integer and k >= 2.

Sequence in context: A277647 A296134 A306694 * A253641 A158378 A052409

Adjacent sequences: A158049 A158050 A158051 * A158053 A158054 A158055

Keyword

sign

Author

Daniel Forgues, Mar 12 2009

Status

approved