

A272027


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


21



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
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OFFSET

1,1


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 nth level of an irregular step pyramid (starting from the top) where the structure of every 120degree threedimensional sector arises after the 120degree zigzag 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 nth level of an irregular step pyramid where the structure of every 360/k threedimensional sector arises after the 360/kdegree zigzag folding of every row of the diagram of the isosceles triangle A237593. If k >= 5 the top of the pyramid is a kpointed 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 120degreezigzag folding (rows: 1..28)
Index entries for sequences related to sigma(n)


FORMULA

a(n) = 3*A000203(n) = A000203(n) + A074400(n) = A239050(n)  A000203(n).
Dirichlet g.f.: 3*zeta(s1)*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


MAPLE

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


MATHEMATICA

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


PROG

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


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.
Sequence in context: A285564 A140979 A096726 * A310323 A212059 A226152
Adjacent sequences: A272024 A272025 A272026 * A272028 A272029 A272030


KEYWORD

nonn,easy


AUTHOR

Omar E. Pol, Apr 18 2016


STATUS

approved



