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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. 41
 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 (list; graph; refs; listen; history; text; internal format)
 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()). - 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 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. 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 Cf. A000005, A277564, A278028. Cf. A000961, A002033, A002182, A007916, A047966, A070941, A108917, A126796, A276024, A326035, A326036, A326037. Sequence in context: A331048 A341593 A288636 * A305253 A294336 A316782 Adjacent sequences:  A089720 A089721 A089722 * A089724 A089725 A089726 KEYWORD easy,nonn AUTHOR Naohiro Nomoto, Jan 07 2004 STATUS approved

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Last modified May 23 23:07 EDT 2022. Contains 353993 sequences. (Running on oeis4.)