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A190902 Product_{ d divides n } d*mu(n/d). 1
1, -2, -3, 0, -5, 36, -7, 0, 0, 100, -11, 0, -13, 196, 225, 0, -17, 0, -19, 0, 441, 484, -23, 0, 0, 676, 0, 0, -29, 810000, -31, 0, 1089, 1156, 1225, 0, -37, 1444, 1521, 0, -41, 3111696, -43, 0, 0, 2116, -47, 0, 0, 0, 2601, 0, -53, 0, 3025, 0, 3249, 3364, -59, 0, -61, 3844 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) is the multiplicative equivalent to Euler's totient function, phi(n) = Sum_{ d divides n } d*mu(n/d).

a(n) = 0 iff n is not squarefree (A013929).

a(n) < 0 iff n is prime (A000040).

a(n) = -n iff n has 1 prime factor.

a(n) = n^(2^(k-1)) iff n has k prime factors (k>1).

LINKS

Table of n, a(n) for n=1..62.

EXAMPLE

a(14) = 1*(1)*2*(-1)*7*(-1)*14*(1) = 14^2 = 196.

MAPLE

with(numtheory): A190902 := proc(n) local d; mul(d*mobius(n/d), d=divisors(n)) end:

A190902 := proc(n) if mobius(n)=0 then 0 elif isprime(n) then -n else n^(2^(nops(factorset(n))-1)) fi end:

MATHEMATICA

a[n_] := Product[d MoebiusMu[n/d], {d, Divisors[n]}];

Array[a, 62] (* Jean-Fran├žois Alcover, Jun 24 2019 *)

PROG

(PARI) a(n)={local(r=1); fordiv(n, d, r*=d*moebius(n/d)); return(r); }

CROSSREFS

Sequence in context: A324640 A122830 A321296 * A115562 A127468 A173720

Adjacent sequences:  A190899 A190900 A190901 * A190903 A190904 A190905

KEYWORD

sign

AUTHOR

Peter Luschny, Jul 22 2011

STATUS

approved

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Last modified May 14 22:40 EDT 2021. Contains 343909 sequences. (Running on oeis4.)