%I
%S 3,9,12,21,18,36,24,45,39,54,36,84,42,72,72,93,54,117,60,126,96,108,
%T 72,180,93,126,120,168,90,216,96,189,144,162,144,273,114,180,168,270,
%U 126,288,132,252,234,216,144,372,171,279,216,294,162,360,216,360,240,270,180,504,186,288,312,381
%N a(n) = 3*sigma(n).
%C 3 times the sum of the divisors of n.
%C From _Omar E. Pol_, Jul 04 2016: (Start)
%C a(n) is also the total number of horizontal rhombuses in the terraces of the nth level of an irregular stepped 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).
%C 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 stepped 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)
%H Antti Karttunen, <a href="/A272027/b272027.txt">Table of n, a(n) for n = 1..10000</a>
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr02.jpg">Diagram of the triangle before the 120degreezigzag folding (rows: 1..28)</a>
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F a(n) = 3*A000203(n) = A000203(n) + A074400(n) = A239050(n)  A000203(n).
%F Dirichlet g.f.: 3*zeta(s1)*zeta(s).  _Ilya Gutkovskiy_, Jul 04 2016
%F a(n) = A274536(n)/2.  _Antti Karttunen_, Nov 16 2017
%F From _Omar E. Pol_, Oct 02 2018: (Start)
%F Conjecture 1: a(n) = sigma(2*n) = A062731(n) iff n is odd.
%F And more generally:
%F Conjecture 2: If p is prime then (p + 1)*sigma(n) = sigma(p*n) iff n is not a multiple of p. (End)
%F 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
%p with(numtheory): seq(3*sigma(n), n=1..64);
%t Table[3 DivisorSigma[1, n], {n, 64}] (* _Michael De Vlieger_, Apr 19 2016 *)
%o (PARI) a(n) = 3 * sigma(n);
%o (MAGMA) [3*SumOfDivisors(n): n in [1..70]]; // _Vincenzo Librandi_, Jul 30 2019
%Y Alternating row sums of triangle A272026.
%Y k times sigma(n), k = 1..10: A000203, A074400, this sequence, A239050, A274535, A274536, A319527, A319528, A325299, A326122.
%Y Cf. A062731, A196020, A236104, A237270, A237593.
%K nonn,easy
%O 1,1
%A _Omar E. Pol_, Apr 18 2016
