A089723 a(1)=1; for n>1, a(n) gives number of ways to write n as n = x^y, 2 <= x, 1 <= y.

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 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, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1

Offset: 1

Author

Naohiro Nomoto, Jan 07 2004

Keywords

• easy
• nonn

Comments

This function depends only on the prime signature of n. - Franklin T. Adams-Watters, Mar 10 2006

a(n) is the number of perfect divisors of n. Perfect divisor of n is divisor d such that d^k = n for some k >= 1. a(n) > 1 for perfect powers n = A001597(m) for m > 2. - Jaroslav Krizek, Jan 23 2010

Also the number of uniform perfect integer partitions of n - 1. An integer partition of n is uniform if all parts appear with the same multiplicity, and perfect if every nonnegative integer up to n is the sum of a unique submultiset. The Heinz numbers of these partitions are given by A326037. The a(16) = 3 partitions are: (8,4,2,1), (4,4,4,1,1,1), (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1). - Gus Wiseman, Jun 07 2019

The record values occur at 1 and at 2^A002182(n) for n > 1. - Amiram Eldar, Nov 06 2020

Examples

			144 = 2^4 * 3^2, gcd(4,2) = 2, d(2) = 2, so a(144) = 2. The representations are 144^1 and 12^2.
From _Friedjof Tellkamp_, Jun 14 2025: (Start)
n:          1, 2, 3, 4, 5, 6, 7, 8, 9, ...
----------------------------------------------------
1st powers: 1, 1, 1, 1, 1, 1, 1, 1, 1, ... (A000012)
Squares:    1, 0, 0, 1, 0, 0, 0, 0, 1, ... (A010052)
Cubes:      1, 0, 0, 0, 0, 0, 0, 1, 0, ... (A010057)
Quartics:   1, 0, 0, 0, 0, 0, 0, 0, 0, ... (A374016)
...
Sum:       oo, 1, 1, 2, 1, 1, 1, 2, 2, ...
a(1)=1:     1, 1, 1, 2, 1, 1, 1, 2, 2, ... (= this sequence). (End)
		

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..20000
• Solomon W. Golomb, A new arithmetic function of combinatorial significance, J. Number Theory, Vol. 5, No. 3 (1973), pp. 218-223. 1973JNT.....5..218G
• Jan Mycielski, Sur les représentations des nombres naturels par des puissances à base et exposant naturels, Colloquium Mathematicum, Vol. 2 (1951), pp. 254-260. See gamma(n).
• N. J. A. Sloane, Maple programs for A007916, A278028, A278029, A052409, A089723, A277564

Crossrefs

Cf. A000005, A277564, A278028.

Cf. A000961, A002033, A002182, A007916, A047966, A070941, A108917, A126796, A276024, A326035, A326036, A326037.

Cf. A010052, A010057, A374016, A259362.

Programs

Maple

with(numtheory):
A089723 := proc(n) local t1,t2,g,j;
if n=1 then 1 else
t1:=ifactors(n)[2]; t2:=nops(t1); g := t1[1][2];
for j from 2 to t2 do g:=gcd(g,t1[j][2]); od:
tau(g); fi; end;
[seq(A089723(n),n=1..100)]; # N. J. A. Sloane, Nov 10 2016

Mathematica

Table[DivisorSigma[0, GCD @@ FactorInteger[n][[All, 2]]], {n, 100}] (* Gus Wiseman, Jun 12 2017 *)

PARI

a(n) = if (n==1, 1, numdiv(gcd(factor(n)[,2]))); \\ Michel Marcus, Jun 13 2017

Python

from math import gcd
from sympy import factorint, divisor_sigma
def a(n):
    if n == 1: return 1
    e = list(factorint(n).values())
    g = e[0]
    for ei in e[1:]: g = gcd(g, ei)
    return divisor_sigma(g, 0)
print([a(n) for n in range(1, 105)]) # Michael S. Branicky, Jul 15 2021

Formula

If n = Product p_i^e_i, a(n) = d(gcd()). - Franklin T. Adams-Watters, Mar 10 2006

Sum_{n=1..m} a(n) = A255165(m) + 1. - Richard R. Forberg, Feb 16 2015

Sum_{n>=2} a(n)/n^s = Sum_{n>=2} 1/(n^s-1) = Sum_{k>=1} (zeta(s*k)-1) for all real s with Re(s) > 1 (Golomb, 1973). - Amiram Eldar, Nov 06 2020

For n > 1, a(n) = Sum_{i=1..floor(n/2)} floor(n^(1/i))-floor((n-1)^(1/i)). - Wesley Ivan Hurt, Dec 08 2020

Sum_{n>=1} (a(n)-1)/n = 1 (Mycielski, 1951). - Amiram Eldar, Jul 15 2021

From Friedjof Tellkamp, Jun 14 2025: (Start)

a(n) = 1 + A259362(n) = 1 + A010052(n) + A010057(n) + A374016(n) + (...), for n > 1.

G.f.: x + Sum_{j>=2, k>=1} x^(j^k). (End)