A290480 Product of proper unitary divisors of n.

1, 1, 1, 1, 1, 6, 1, 1, 1, 10, 1, 12, 1, 14, 15, 1, 1, 18, 1, 20, 21, 22, 1, 24, 1, 26, 1, 28, 1, 27000, 1, 1, 33, 34, 35, 36, 1, 38, 39, 40, 1, 74088, 1, 44, 45, 46, 1, 48, 1, 50, 51, 52, 1, 54, 55, 56, 57, 58, 1, 216000, 1, 62, 63, 1, 65, 287496, 1, 68, 69, 343000, 1, 72, 1, 74, 75, 76, 77, 474552, 1, 80

Offset

1,6

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

a(n) = A061537(n)/n.

a(n) = n^(2^(omega(n)-1)-1), where omega() is the number of distinct primes dividing n (A001221).

a(n) = 1 if n is a prime power.

Example

a(12) = 12 because 12 has 6 divisors {1, 2, 3, 4, 6, 12} among which 3 are proper unitary {1, 3, 4} and 1*3*4 = 12.

Maple

with(numtheory):
a:= n-> mul(d, d=select(x-> igcd(x, n/x)=1, divisors(n) minus {n})):
seq(a(n), n=1..80);  # Alois P. Heinz, Aug 03 2017

Mathematica

Table[Product[d, {d, Select[Divisors[n], GCD[#, n/#] == 1 &]}]/n, {n, 80}]
Table[n^(2^(PrimeNu[n] - 1) - 1), {n, 80}]

Prog

(Python)
from sympy import divisors, gcd, prod
def a(n): return prod(d for d in divisors(n) if gcd(d, n//d) == 1)//n
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Aug 04 2017
(PARI) A290480(n) = if(1==n, n, n^(2^(omega(n)-1)-1)); \\ Antti Karttunen, Aug 06 2018

Crossrefs

Cf. A001221, A007774 (fixed points), A007955, A007956, A034460, A061537, A062758, A063919, A078599, A087652, A126192, A136655, A183091.

Sequence in context: A166142 A290479 A304404 * A183092 A050449 A316623

Adjacent sequences: A290477 A290478 A290479 * A290481 A290482 A290483

Keyword

nonn

Author

Ilya Gutkovskiy, Aug 03 2017

Status

approved