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

%I A089723 #86 Jun 14 2025 14:55:13
%S A089723 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,
%T A089723 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,
%U A089723 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
%N A089723 a(1)=1; for n>1, a(n) gives number of ways to write n as n = x^y, 2 <= x, 1 <= y.
%C A089723 This function depends only on the prime signature of n. - _Franklin T. Adams-Watters_, Mar 10 2006
%C A089723 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
%C A089723 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
%C A089723 The record values occur at 1 and at 2^A002182(n) for n > 1. - _Amiram Eldar_, Nov 06 2020
%H A089723 N. J. A. Sloane, <a href="/A089723/b089723.txt">Table of n, a(n) for n = 1..20000</a>
%H A089723 Solomon W. Golomb, <a href="http://dx.doi.org/10.1016/0022-314X(73)90047-4">A new arithmetic function of combinatorial significance</a>, J. Number Theory, Vol. 5, No. 3 (1973), pp. 218-223. <a href="http://adsabs.harvard.edu/abs/1973JNT.....5..218G">1973JNT.....5..218G</a>
%H A089723 Jan Mycielski, <a href="http://matwbn.icm.edu.pl/ksiazki/cm/cm2/cm2140.pdf">Sur les représentations des nombres naturels par des puissances à base et exposant naturels</a>, Colloquium Mathematicum, Vol. 2 (1951), pp. 254-260. See gamma(n).
%H A089723 N. J. A. Sloane, <a href="/A278028/a278028.txt">Maple programs for A007916, A278028, A278029, A052409, A089723, A277564</a>
%F A089723 If n = Product p_i^e_i, a(n) = d(gcd(<e_i>)). - _Franklin T. Adams-Watters_, Mar 10 2006
%F A089723 Sum_{n=1..m} a(n) = A255165(m) + 1. - _Richard R. Forberg_, Feb 16 2015
%F A089723 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
%F A089723 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
%F A089723 Sum_{n>=1} (a(n)-1)/n = 1 (Mycielski, 1951). - _Amiram Eldar_, Jul 15 2021
%F A089723 From _Friedjof Tellkamp_, Jun 14 2025: (Start)
%F A089723 a(n) = 1 + A259362(n) = 1 + A010052(n) + A010057(n) + A374016(n) + (...), for n > 1.
%F A089723 G.f.: x + Sum_{j>=2, k>=1} x^(j^k). (End)
%e A089723 144 = 2^4 * 3^2, gcd(4,2) = 2, d(2) = 2, so a(144) = 2. The representations are 144^1 and 12^2.
%e A089723 From _Friedjof Tellkamp_, Jun 14 2025: (Start)
%e A089723 n:          1, 2, 3, 4, 5, 6, 7, 8, 9, ...
%e A089723 ----------------------------------------------------
%e A089723 1st powers: 1, 1, 1, 1, 1, 1, 1, 1, 1, ... (A000012)
%e A089723 Squares:    1, 0, 0, 1, 0, 0, 0, 0, 1, ... (A010052)
%e A089723 Cubes:      1, 0, 0, 0, 0, 0, 0, 1, 0, ... (A010057)
%e A089723 Quartics:   1, 0, 0, 0, 0, 0, 0, 0, 0, ... (A374016)
%e A089723 ...
%e A089723 Sum:       oo, 1, 1, 2, 1, 1, 1, 2, 2, ...
%e A089723 a(1)=1:     1, 1, 1, 2, 1, 1, 1, 2, 2, ... (= this sequence). (End)
%p A089723 with(numtheory):
%p A089723 A089723 := proc(n) local t1,t2,g,j;
%p A089723 if n=1 then 1 else
%p A089723 t1:=ifactors(n)[2]; t2:=nops(t1); g := t1[1][2];
%p A089723 for j from 2 to t2 do g:=gcd(g,t1[j][2]); od:
%p A089723 tau(g); fi; end;
%p A089723 [seq(A089723(n),n=1..100)]; # _N. J. A. Sloane_, Nov 10 2016
%t A089723 Table[DivisorSigma[0, GCD @@ FactorInteger[n][[All, 2]]], {n, 100}] (* _Gus Wiseman_, Jun 12 2017 *)
%o A089723 (PARI) a(n) = if (n==1, 1, numdiv(gcd(factor(n)[,2]))); \\ _Michel Marcus_, Jun 13 2017
%o A089723 (Python)
%o A089723 from math import gcd
%o A089723 from sympy import factorint, divisor_sigma
%o A089723 def a(n):
%o A089723     if n == 1: return 1
%o A089723     e = list(factorint(n).values())
%o A089723     g = e[0]
%o A089723     for ei in e[1:]: g = gcd(g, ei)
%o A089723     return divisor_sigma(g, 0)
%o A089723 print([a(n) for n in range(1, 105)]) # _Michael S. Branicky_, Jul 15 2021
%Y A089723 Cf. A000005, A277564, A278028.
%Y A089723 Cf. A000961, A002033, A002182, A007916, A047966, A070941, A108917, A126796, A276024, A326035, A326036, A326037.
%Y A089723 Cf. A010052, A010057, A374016, A259362.
%K A089723 easy,nonn
%O A089723 1,4
%A A089723 _Naohiro Nomoto_, Jan 07 2004