A259362 - OEIS

A259362

a(1) = 1, for n > 1: a(n) is the number of ways to write n as a nontrivial perfect power.

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0

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OFFSET

1,16

COMMENTS

a(n) = number of integer pairs (i,j) for distinct values of i where i > 0, j > 1 and n = i^j. Since 1 = 1^r for all real values of r, the requirement for a distinct i causes a(1) = 1 instead of a(1) = infinity.

Alternatively, the sequence can be defined as: a(1) = 1, for n > 1: a(n) = number of pairs (i,j) such that i > 0, j > 1 and n = i^j.

A007916 = n, where a(n) = 0.

A001597 = n, where a(n) > 0.

A175082 = n, where n = 1 or a(n) = 0.

A117453 = n, where n = 1 or a(n) > 1.

A175065 = n, where n > 1 and a(n) > 0 and this is the first occurrence in this sequence of a(n).

A072103 = n repeated a(n) times where n > 1.

A075802 = min(1, a(n)).

A175066 = a(n), where n = 1 or a(n) > 1. This sequence is an expansion of A175066.

A253642 = 0 followed by a(n), where n > 1 and a(n) > 0.

A175064 = a(1) followed by a(n) + 1, where n > 1 and a(n) > 0.

Where n > 1, A001597(x) = n (which implies a(n) > 0), i = A025478(x) and j = A253641(n), then a(n) = A000005(j) - 1, which is the number of factors of j greater than 1. The integer pair (i,j) comprises the smallest value i and the largest value j where i > 0, j > 1 and n = i^j. The a(n) pairs of (a,b) where a > 0, b > 1 and n = a^b are formed with b = each of the a(n) factors of j greater than 1. Examples for n = {8,4096}:

a(8) = 1, A001597(3) = 8, A025478(3) = 2, A253641(8) = 3, 8 = 2^3 and A000005(3) - 1 = 1 because there is one factor of 3 greater than 1 [3]. The set of pairs (a,b) is {(2,3)}.

a(4096) = 5, A001597(82) = 4096, A025478(82) = 2, A253641(4096) = 12, 4096 = 2^12 and A000005(12) - 1 = 5 because there are five factors of 12 greater than 1 [2,3,4,6,12]. The set of pairs (a,b) is {(64,2),(16,3),(8,4),(4,6),(2,12)}.

A023055 = the ordered list of x+1 with duplicates removed, where x is the number of consecutive zeros appearing in this sequence between any two nonzero terms.

A070428(x) = number of terms a(n) > 0 where n <= 10^x.

a(n) <= A188585(n).

LINKS

Doug Bell, Table of n, a(n) for n = 1..5000

FORMULA

a(1) = 1, for n > 1: a(n) = A000005(A253641(n)) - 1.

If n not in A001597, then a(n) = 0, otherwise a(n) = A175064(x) - 1 where A001597(x) = n.

EXAMPLE

a(6) = 0 because there is no way to write 6 as a nontrivial perfect power.

a(9) = 1 because there is one way to write 9 as a nontrivial perfect power: 3^2.

a(16) = 2 because there are two ways to write 16 as a nontrivial perfect power: 2^4, 4^2.

CROSSREFS