login
Denominator of (n-d)/n*d where d = A000005(n) is the number of divisors of n.
3

%I #16 Apr 18 2022 04:12:41

%S 1,1,6,12,10,12,14,8,9,20,22,12,26,28,60,80,34,9,38,60,84,44,46,12,75,

%T 52,108,84,58,120,62,96,132,68,140,12,74,76,156,10,82,168,86,132,90,

%U 92,94,240,147,75,204,156,106,216,220,28,228,116,118,15,122,124,126,448,260,264,134

%N Denominator of (n-d)/n*d where d = A000005(n) is the number of divisors of n.

%C The terms are of course the denominators of the fraction "in smallest terms", otherwise said: a(n) = n*d/gcd(n*d, n - d), which is unambiguous also for n = 1 and n = 2 where n - d = 0.

%H M. F. Hasler, <a href="/A352482/b352482.txt">Table of n, a(n) for n = 1..10000</a> (terms a(3..10^4) from Michel Marcus), Apr 17 2022

%e The number n = 1 has d = 1 divisors, so (n-d)/(n*d) = 0/1 has denominator a(1) = 1.

%e The number n = 2 has d = 2 divisors, so (n-d)/(n*d) = 0/4 = 0/1 has denominator a(2) = 1 when written in smallest terms.

%e The number n = 3 has d = 2 divisors, so (n-d)/(n*d) = 1/6 has denominator a(3) = 6.

%e The number n = 4 has d = 3 divisors, so (n-d)/(n*d) = 1/12 has denominator a(4) = 12.

%e The number n = 6 has d = 4 divisors, so (n-d)/(n*d) = 2/24 = 1/12 has denominator a(6) = 12.

%t a[n_] := Numerator[n*(d = DivisorSigma[0, n])/(n - d)]; Array[a, 100, 3] (* _Amiram Eldar_, Mar 18 2022 *)

%o (PARI) A352482(n,d=numdiv(n))=denominator((n-d)/(n*d))

%Y Cf. A000005, A049820, A146566, A352483 (numerator).

%K nonn,frac

%O 1,3

%A _Michel Marcus_, Mar 18 2022

%E Edited and extended to offset 1 by _M. F. Hasler_, Apr 17 2022