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A239050 a(n) = 4*sigma(n). 29

%I #83 Sep 08 2022 08:46:07

%S 4,12,16,28,24,48,32,60,52,72,48,112,56,96,96,124,72,156,80,168,128,

%T 144,96,240,124,168,160,224,120,288,128,252,192,216,192,364,152,240,

%U 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

%N a(n) = 4*sigma(n).

%C 4 times the sum of divisors of n.

%C 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

%H Antti Karttunen, <a href="/A239050/b239050.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 90-degree-zig-zag folding (rows: 1..28)</a>

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr01.jpg">Folding the first eight rows of triangle</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

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

%F a(n) = A000203(n) + A272027(n). - _Omar E. Pol_, Jul 04 2016

%F Dirichlet g.f.: 4*zeta(s-1)*zeta(s). - _Ilya Gutkovskiy_, Jul 04 2016

%F Conjecture: a(n) = sigma(3*n) = A144613(n) iff n is not a multiple of 3. - _Omar E. Pol_, Oct 02 2018

%F 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

%e For n = 4 the sum of divisors of 4 is 1 + 2 + 4 = 7, so a(4) = 4*7 = 28.

%e For n = 5 the sum of divisors of 5 is 1 + 5 = 6, so a(5) = 4*6 = 24.

%e .

%e Illustration of initial terms: _ _ _ _ _ _

%e . _ _ _ _ _ _ |_|_|_|_|_|_|

%e . _ _ _ _ _|_|_|_|_|_|_|_ _ _| |_ _

%e . _ _ _ _ _|_|_|_|_|_ |_|_| |_|_| |_| |_|

%e . _ _ |_|_|_|_| |_| |_| |_| |_| |_| |_|

%e . |_|_| |_| |_| |_| |_| |_| |_| |_| |_|

%e . |_|_| |_|_ _|_| |_| |_| |_| |_| |_| |_|

%e . |_|_|_|_| |_|_ _ _ _|_| |_|_ _|_| |_| |_|

%e . |_|_|_|_| |_|_|_ _ _ _|_|_| |_|_ _|_|

%e . |_|_|_|_|_|_| |_ _ _ _ _ _|

%e . |_|_|_|_|_|_|

%e .

%e n: 1 2 3 4 5

%e S(n): 1 3 4 7 6

%e a(n): 4 12 16 28 24

%e .

%e 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.

%e 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

%p with(numtheory): seq(4*sigma(n), n=1..64); # _Omar E. Pol_, Jul 04 2016

%t Array[4 DivisorSigma[1, #] &, 64] (* _Michael De Vlieger_, Nov 16 2017 *)

%o (PARI) a(n) = 4 * sigma(n); \\ _Omar E. Pol_, Jul 04 2016

%o (Magma) [4*SumOfDivisors(n): n in [1..70]]; // _Vincenzo Librandi_, Jul 30 2019

%Y Alternating row sums of A239662.

%Y Partial sums give A243980.

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

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

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

%K nonn,easy

%O 1,1

%A _Omar E. Pol_, Mar 09 2014

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Last modified March 29 01:36 EDT 2024. Contains 371264 sequences. (Running on oeis4.)