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 A272027 a(n) = 3*sigma(n). 22

%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 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 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 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 120-degree-zig-zag 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(s-1)*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

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Last modified April 10 11:33 EDT 2021. Contains 342845 sequences. (Running on oeis4.)