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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
sign
AUTHOR
Daniel Forgues, Mar 12 2009
STATUS
approved

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Last modified March 28 17:42 EDT 2024. Contains 371254 sequences. (Running on oeis4.)