A239050 a(n) = 4*sigma(n).

4, 12, 16, 28, 24, 48, 32, 60, 52, 72, 48, 112, 56, 96, 96, 124, 72, 156, 80, 168, 128, 144, 96, 240, 124, 168, 160, 224, 120, 288, 128, 252, 192, 216, 192, 364, 152, 240, 224, 360, 168, 384, 176, 336, 312, 288, 192, 496, 228, 372, 288, 392, 216, 480, 288, 480, 320, 360, 240, 672, 248, 384, 416, 508

Offset: 1

Author

Omar E. Pol, Mar 09 2014

Keywords

• nonn
• easy

Comments

4 times the sum of divisors of n.

a(n) is also the total number of horizontal cells in the terraces of the n-th level of an irregular stepped pyramid (starting from the top) where the structure of every three-dimensional quadrant arises after the 90-degree zig-zag folding of every row of the diagram of the isosceles triangle A237593. The top of the pyramid is a square formed by four cells (see links and examples). - Omar E. Pol, Jul 04 2016

Examples

			For n = 4 the sum of divisors of 4 is 1 + 2 + 4 = 7, so a(4) = 4*7 = 28.
For n = 5 the sum of divisors of 5 is 1 + 5 = 6, so a(5) = 4*6 = 24.
.
Illustration of initial terms:                                    _ _ _ _ _ _
.                                           _ _ _ _ _ _          |_|_|_|_|_|_|
.                           _ _ _ _       _|_|_|_|_|_|_|_     _ _|           |_ _
.             _ _ _ _     _|_|_|_|_|_    |_|_|       |_|_|   |_|               |_|
.     _ _    |_|_|_|_|   |_|       |_|   |_|           |_|   |_|               |_|
.    |_|_|   |_|   |_|   |_|       |_|   |_|           |_|   |_|               |_|
.    |_|_|   |_|_ _|_|   |_|       |_|   |_|           |_|   |_|               |_|
.            |_|_|_|_|   |_|_ _ _ _|_|   |_|_         _|_|   |_|               |_|
.                          |_|_|_|_|     |_|_|_ _ _ _|_|_|   |_|_             _|_|
.                                          |_|_|_|_|_|_|         |_ _ _ _ _ _|
.                                                                |_|_|_|_|_|_|
.
n:     1          2             3                4                     5
S(n):  1          3             4                7                     6
a(n):  4         12            16               28                    24
.
For n = 1..5, the figure n represents the reflection in the four quadrants of the symmetric representation of S(n) = sigma(n) = A000203(n). For more information see A237270 and A237593.
The diagram also represents the top view of the first four terraces of the stepped pyramid described in Comments section. - _Omar E. Pol_, Jul 04 2016
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Omar E. Pol, Diagram of the triangle before the 90-degree-zig-zag folding (rows: 1..28)
• Omar E. Pol, Folding the first eight rows of triangle
• Index entries for sequences related to sigma(n)

Crossrefs

Alternating row sums of A239662.

Partial sums give A243980.

k times sigma(n), k=1..6: A000203, A074400, A272027, this sequence, A274535, A274536.

k times sigma(n), k = 1..10: A000203, A074400, A272027, this sequence, A274535, A274536, A319527, A319528, A325299, A326122.

Cf. A008438, A017113, A062731, A112610, A144613, A193553, A196020, A235791, A236104, A237270, A237593, A239052, A239053, A239660, A239662, A244050, A262626.

Programs

Magma

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

Maple

with(numtheory): seq(4*sigma(n), n=1..64); # Omar E. Pol, Jul 04 2016

Mathematica

Array[4 DivisorSigma[1, #] &, 64] (* Michael De Vlieger, Nov 16 2017 *)

PARI

a(n) = 4 * sigma(n); \\ Omar E. Pol, Jul 04 2016

Formula

a(n) = 4*A000203(n) = 2*A074400(n).

a(n) = A000203(n) + A272027(n). - Omar E. Pol, Jul 04 2016

Dirichlet g.f.: 4*zeta(s-1)*zeta(s). - Ilya Gutkovskiy, Jul 04 2016

Conjecture: a(n) = sigma(3*n) = A144613(n) iff n is not a multiple of 3. - Omar E. Pol, Oct 02 2018

The conjecture above is correct. Write n = 3^e*m, gcd(3, m) = 1, then sigma(3*n) = sigma(3^(e+1))*sigma(m) = ((3^(e+2) - 1)/2)*sigma(m) = ((3^(e+2) - 1)/(3^(e+1) - 1))*sigma(3^e*m), and (3^(e+2) - 1)/(3^(e+1) - 1) = 4 if and only if e = 0. - Jianing Song, Feb 03 2019