A265390 a(n) = lcm_{d|n} tau(d) * Sum_{d|n} 1/tau(d), where tau(d) represents the number of divisors of d (A000005(d)).

1, 3, 3, 11, 3, 9, 3, 25, 11, 9, 3, 33, 3, 9, 9, 137, 3, 33, 3, 33, 9, 9, 3, 75, 11, 9, 25, 33, 3, 27, 3, 147, 9, 9, 9, 121, 3, 9, 9, 75, 3, 27, 3, 33, 33, 9, 3, 411, 11, 33, 9, 33, 3, 75, 9, 75, 9, 9, 3, 99, 3, 9, 33, 1089, 9, 27, 3, 33, 9, 27, 3, 275, 3, 9, 33, 33, 9, 27, 3, 411, 137, 9, 3, 99, 9, 9, 9, 75, 3, 99, 9, 33

Offset: 1

Author

Jaroslav Krizek, Dec 08 2015

Keywords

• nonn
• mult

Examples

			For n = 6; divisors d of 6: {1, 2, 3, 6}; tau(d): {1, 2, 2, 4}; LCM_{d|6} tau(d) = 4; a(6) = 4/1 + 4/2 + 4/2 + 4/4 = 9.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A000005, A025529, A253139, A265391, A265392, A265393.

Programs

Magma

[&+[LCM([NumberOfDivisors(d): d in Divisors(n)]) / NumberOfDivisors(d): d in Divisors(n) ]: n in [1..100]]

Mathematica

Table[LCM @@ DivisorSigma[0, Divisors@ n] Sum[1/DivisorSigma[0, d], {d, Divisors@ n}], {n, 74}] (* Michael De Vlieger, Dec 09 2015 *)

PARI

A253139(n) = my(d = divisors(n)); lcm(vector(#d, k, numdiv(d[k]))); \\ This function from Michel Marcus, Jan 23 2015
A265390(n) = (A253139(n) * sumdiv(n,d,(1/numdiv(d)))); \\ Antti Karttunen, Nov 24 2017

Formula

a(n) = A253139(n) * Sum_{d|n} 1/A000005(d) = A265391(n) * A253139(n) / A265392(n).

Multiplicative with a(p^e) = A025529(e+1) = (1/1 + 1/2 + 1/3 + ... + 1/(e+1)) * lcm{1, 2, 3, ..., e+1}.

Extensions

More terms from Antti Karttunen, Nov 24 2017