A323707 a(n) = denominator of Sum_{d|n} tau(d)/pod(d) where tau(k) = the number of the divisors of k (A000005) and pod(k) = the product of the divisors of k (A007955).

1, 1, 3, 8, 5, 9, 7, 16, 9, 25, 11, 32, 13, 49, 225, 1024, 17, 972, 19, 4000, 441, 121, 23, 41472, 125, 169, 729, 10976, 29, 101250, 31, 16384, 1089, 289, 1225, 1119744, 37, 361, 1521, 320000, 41, 388962, 43, 42592, 30375, 529, 47, 127401984, 343, 62500, 2601

Offset

1,3

Comments

Sum_{d|n} tau(d)/pod(d) > 1 for all n > 1.

Links

Table of n, a(n) for n=1..51.

Formula

a(n) = 1 for n = 1, 2, ... (no other n <= 5*10^6).

a(p) = p for prime p > 2.

Example

For n=4; Sum_{d|4} tau(d)/pod(d) = tau(1)/pod(1) + tau(2)/pod(2) + tau(4)/pod(4) = 1/1 + 2/2 + 3/8 = 19/8; a(4) = 8.

Mathematica

Array[Denominator@ DivisorSum[#, DivisorSigma[0, #]/Apply[Times, Divisors@ #] &] &, 51] (* Michael De Vlieger, Jan 27 2019 *)

Prog

(Magma) [Denominator(&+[NumberOfDivisors(d) / &*[c: c in Divisors(d)]: d in Divisors(n)]): n in [1..100]]
(PARI) a(n) = denominator(sumdiv(n, d, numdiv(d)/vecprod(divisors(d)))); \\ Michel Marcus, Jan 26 2019

Crossrefs

Cf. A000005, A007955, A323706 (numerator).

Sequence in context: A375596 A342934 A256616 * A291186 A347942 A058055

Adjacent sequences: A323704 A323705 A323706 * A323708 A323709 A323710

Keyword

nonn,frac

Author

Jaroslav Krizek, Jan 26 2019

Status

approved