OFFSET
1,1
COMMENTS
4 times the sum of divisors of n.
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
LINKS
FORMULA
Dirichlet g.f.: 4*zeta(s-1)*zeta(s). - Ilya Gutkovskiy, Jul 04 2016
Conjecture: a(n) = sigma(3*n) = A144613(n) iff n is not a multiple of 3. - Omar E. Pol, Oct 02 2018
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
EXAMPLE
For n = 4 the sum of divisors of 4 is 1 + 2 + 4 = 7, so a(4) = 4*7 = 28.
For n = 5 the sum of divisors of 5 is 1 + 5 = 6, so a(5) = 4*6 = 24.
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Illustration of initial terms: _ _ _ _ _ _
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n: 1 2 3 4 5
S(n): 1 3 4 7 6
a(n): 4 12 16 28 24
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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.
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
MAPLE
with(numtheory): seq(4*sigma(n), n=1..64); # Omar E. Pol, Jul 04 2016
MATHEMATICA
Array[4 DivisorSigma[1, #] &, 64] (* Michael De Vlieger, Nov 16 2017 *)
PROG
(PARI) a(n) = 4 * sigma(n); \\ Omar E. Pol, Jul 04 2016
(Magma) [4*SumOfDivisors(n): n in [1..70]]; // Vincenzo Librandi, Jul 30 2019
CROSSREFS
Alternating row sums of A239662.
Partial sums give A243980.
k times sigma(n), k = 1..10: A000203, A074400, A272027, this sequence, A274535, A274536, A319527, A319528, A325299, A326122.
KEYWORD
nonn,easy
AUTHOR
Omar E. Pol, Mar 09 2014
STATUS
approved