OFFSET
1,4
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
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).
FORMULA
If n = Product p_i^e_i, a(n) = d(gcd(<e_i>)). - 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)
G.f.: x + Sum_{j>=2, k>=1} x^(j^k). (End)
EXAMPLE
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)
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 *)
PROG
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Naohiro Nomoto, Jan 07 2004
STATUS
approved