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

Offset: 1

Author

Doug Bell, Jun 24 2015

Keywords

• nonn

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).

Examples

			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.
From _Friedjof Tellkamp_, Jun 14 2025: (Start)
n:       1, 2, 3, 4, 5, 6, 7, 8, 9, ...
Squares: 1, 0, 0, 1, 0, 0, 0, 0, 1, ... (A010052)
Cubes:   1, 0, 0, 0, 0, 0, 0, 1, 0, ... (A010057)
...
Sum:    oo, 0, 0, 1, 0, 0, 0, 1, 1, ...
a(1)=1:  1, 0, 0, 1, 0, 0, 0, 1, 1, ... (= this sequence). (End)
		

Links

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

Crossrefs

Cf. A001597, A007916, A023055, A025478, A070428, A072103, A075090, A075109, A075802, A117453, A175064, A175066, A175082, A188585, A253641, A253642.

Cf. A010052, A010057, A374016, A089361, A089723, A382691 (alternating).

Programs

Mathematica

a[n_] := If[n == 1, 1, Sum[Boole[IntegerQ[n^(1/k)]], {k, 2, Floor[Log[2, n]]}]]; Array[a, 100] (* Friedjof Tellkamp, Jun 14 2025 *)
a[n_] := If[n == 1, 1, DivisorSigma[0, Apply[GCD, Transpose[FactorInteger[n]][[2]]]] - 1]; Array[a, 100] (* Michael Shamos, Jul 06 2025 *)

PARI

a(n) = if (n==1, 1, sum(i=2, logint(n, 2), ispower(n, i))); \\ Michel Marcus, Apr 11 2025

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.

From Friedjof Tellkamp, Jun 14 2025: (Start)

a(n) = A089723(n) - 1, for n > 1.

a(n) = A010052(n) + A010057(n) + A374016(n) + (...), for n > 1.

Sum_{k>=2..n} a(k) = A089361(n), for n > 1.

G.f.: x + Sum_{j>=2, k>=2} x^(j^k).

Dirichlet g.f.: 1 + Sum_{k>=2} zeta(k*s)-1. (End)