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

%I A272027 #85 Sep 08 2022 08:46:16
%S A272027 3,9,12,21,18,36,24,45,39,54,36,84,42,72,72,93,54,117,60,126,96,108,
%T A272027 72,180,93,126,120,168,90,216,96,189,144,162,144,273,114,180,168,270,
%U A272027 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 A272027 a(n) = 3*sigma(n).
%C A272027 3 times the sum of the divisors of n.
%C A272027 From _Omar E. Pol_, Jul 04 2016: (Start)
%C A272027 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).
%C A272027 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)
%H A272027 Antti Karttunen, <a href="/A272027/b272027.txt">Table of n, a(n) for n = 1..10000</a>
%H A272027 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr02.jpg">Diagram of the triangle before the 120-degree-zig-zag folding (rows: 1..28)</a>
%H A272027 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A272027 a(n) = 3*A000203(n) = A000203(n) + A074400(n) = A239050(n) - A000203(n).
%F A272027 Dirichlet g.f.: 3*zeta(s-1)*zeta(s). - _Ilya Gutkovskiy_, Jul 04 2016
%F A272027 a(n) = A274536(n)/2. - _Antti Karttunen_, Nov 16 2017
%F A272027 From _Omar E. Pol_, Oct 02 2018: (Start)
%F A272027 Conjecture 1: a(n) = sigma(2*n) = A062731(n) iff n is odd.
%F A272027 And more generally:
%F A272027 Conjecture 2: If p is prime then (p + 1)*sigma(n) = sigma(p*n) iff n is not a multiple of p. (End)
%F A272027 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 A272027 with(numtheory): seq(3*sigma(n), n=1..64);
%t A272027 Table[3 DivisorSigma[1, n], {n, 64}] (* _Michael De Vlieger_, Apr 19 2016 *)
%o A272027 (PARI) a(n) = 3 * sigma(n);
%o A272027 (Magma) [3*SumOfDivisors(n): n in [1..70]]; // _Vincenzo Librandi_, Jul 30 2019
%Y A272027 Alternating row sums of triangle A272026.
%Y A272027 k times sigma(n), k = 1..10: A000203, A074400, this sequence, A239050, A274535, A274536, A319527, A319528, A325299, A326122.
%Y A272027 Cf. A062731, A196020, A236104, A237270, A237593.
%K A272027 nonn,easy
%O A272027 1,1
%A A272027 _Omar E. Pol_, Apr 18 2016