A272027 a(n) = 3*sigma(n).

3, 9, 12, 21, 18, 36, 24, 45, 39, 54, 36, 84, 42, 72, 72, 93, 54, 117, 60, 126, 96, 108, 72, 180, 93, 126, 120, 168, 90, 216, 96, 189, 144, 162, 144, 273, 114, 180, 168, 270, 126, 288, 132, 252, 234, 216, 144, 372, 171, 279, 216, 294, 162, 360, 216, 360, 240, 270, 180, 504, 186, 288, 312, 381

Offset: 1

Author

Omar E. Pol, Apr 18 2016

Keywords

• nonn
• easy

Comments

3 times the sum of the divisors of n.

From Omar E. Pol, Jul 04 2016: (Start)

a(n) is also the total number of horizontal rhombuses in the terraces of the n-th level of an irregular stepped pyramid (starting from the top) where the structure of every 120-degree three-dimensional sector arises after the 120-degree zig-zag folding of every row of the diagram of the isosceles triangle A237593. The top of the pyramid is a hexagon formed by three rhombuses (see Links section).

More generally, if k >= 3 then k*sigma(n) is also the total number of horizontal rhombuses in the terraces of the n-th level of an irregular stepped pyramid where the structure of every 360/k three-dimensional sector arises after the 360/k-degree zig-zag folding of every row of the diagram of the isosceles triangle A237593. If k >= 5 the top of the pyramid is a k-pointed star formed by k rhombuses. (End)

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Omar E. Pol, Diagram of the triangle before the 120-degree-zig-zag folding (rows: 1..28)
• Index entries for sequences related to sigma(n)

Crossrefs

Alternating row sums of triangle A272026.

k times sigma(n), k = 1..10: A000203, A074400, this sequence, A239050, A274535, A274536, A319527, A319528, A325299, A326122.

Cf. A062731, A196020, A236104, A237270, A237593.

Programs

Magma

[3*SumOfDivisors(n): n in [1..70]]; // Vincenzo Librandi, Jul 30 2019

Maple

with(numtheory): seq(3*sigma(n), n=1..64);

Mathematica

Table[3 DivisorSigma[1, n], {n, 64}] (* Michael De Vlieger, Apr 19 2016 *)

PARI

a(n) = 3 * sigma(n);

Formula

a(n) = 3*A000203(n) = A000203(n) + A074400(n) = A239050(n) - A000203(n).

Dirichlet g.f.: 3*zeta(s-1)*zeta(s). - Ilya Gutkovskiy, Jul 04 2016

a(n) = A274536(n)/2. - Antti Karttunen, Nov 16 2017

From Omar E. Pol, Oct 02 2018: (Start)

Conjecture 1: a(n) = sigma(2*n) = A062731(n) iff n is odd.

And more generally:

Conjecture 2: If p is prime then (p + 1)*sigma(n) = sigma(p*n) iff n is not a multiple of p. (End)

The above claims easily follow from the fact that sigma is multiplicative function, thus if p does not divide n, then sigma(p*n) = sigma(p)*sigma(n). - Antti Karttunen, Nov 21 2019