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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. 33
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) = 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

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

S. W. Golomb, A new arithmetic function of combinatorial significance, J. Number Theory 5 (1973) 218-223. 1973JNT.....5..218G

N. J. A. Sloane, Maple programs for A007916, A278028, A278029, A052409, A089723, A277564

FORMULA

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

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

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

CROSSREFS

Cf. A000005, A277564, A278028.

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

Sequence in context: A086074 A180601 A288636 * A305253 A294336 A316782

Adjacent sequences:  A089720 A089721 A089722 * A089724 A089725 A089726

KEYWORD

easy,nonn,changed

AUTHOR

Naohiro Nomoto, Jan 07 2004

STATUS

approved

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Last modified June 17 15:25 EDT 2019. Contains 324194 sequences. (Running on oeis4.)