A190902 a(n) = Product_{d|n} d*mu(n/d).

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

Offset

1,2

Comments

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

a(n) is the determinant of the symmetric tau(n) X tau(n) matrix M(n) defined by M(n) = [sigma(d_i*d_j)], where d_i, d_j run through the divisors of n. - Ridouane Oudra, Apr 25 2026

Links

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

Formula

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

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

a(n) = -n iff n is prime.

From Ridouane Oudra, Apr 25 2026: (Start)

a(n) = A069158(n)*A007955(n).

a(n) = A069158(n)*A061537(n).

a(n) = n^(tau(n)/2) for n in A000469.

a(n) = (A076479(n)*n)^(tau(n)/2) for n in A005117. (End)

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:
# Alternative:
with(numtheory): 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-François Alcover, Jun 24 2019 *)
(* Alternative: *)
a[n_]:=Det[Table[DivisorSigma[1, Part[d=Divisors[n], i]*Part[d, j]], {i, DivisorSigma[0, n]}, {j, DivisorSigma[0, n]}]]; Array[a, 62] (* Stefano Spezia, Apr 26 2026 *)

Prog

(PARI) a(n)={my(r=1); fordiv(n, d, r*=d*moebius(n/d)); return(r); }
(Python)
from sympy import mobius, primenu
def A190902(n): return 0 if not mobius(n) else 1 if n==1 else -n**(1<<m-1) if (m:=primenu(n))==1 else n**(1<<m-1) # Chai Wah Wu, Apr 25 2026

Crossrefs

Cf. A008683, A000005, A000010, A000203, A007955, A061537, A001221, A069158, A000469, A005117, A076479.

Sequence in context: A350260 A358276 A321296 * A115562 A343069 A127468

Adjacent sequences: A190899 A190900 A190901 * A190903 A190904 A190905

Keyword

sign

Author

Peter Luschny, Jul 22 2011

Status

approved