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A327329
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Twice the sum of all divisors of all positive integers <= n.
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8
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2, 8, 16, 30, 42, 66, 82, 112, 138, 174, 198, 254, 282, 330, 378, 440, 476, 554, 594, 678, 742, 814, 862, 982, 1044, 1128, 1208, 1320, 1380, 1524, 1588, 1714, 1810, 1918, 2014, 2196, 2272, 2392, 2504, 2684, 2768, 2960, 3048, 3216, 3372, 3516, 3612, 3860, 3974, 4160, 4304, 4500, 4608, 4848, 4992
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OFFSET
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1,1
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COMMENTS
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a(n) has a symmetric representation. Using two opposite quadrants, where in each quadrant there is the Dyck path related to partitions described in the n-th row of triangle A237593, a(n) is the total area (or the total number of cells) of the structure (see the example).
a(n) is also the total area of the horizontal faces in the stepped pyramid with n levels described in A245092 (that is the total area of the terraces plus the area of the base). - Omar E. Pol, Dec 15 2021
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LINKS
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FORMULA
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EXAMPLE
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Illustration of a(8) = 112 using a symmetric structure constructed with the Dyck path related to partitions described in the 8th row of triangle A237593.
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MATHEMATICA
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Accumulate[2*DivisorSigma[1, Range[60]]] (* Harvey P. Dale, Sep 25 2021 *)
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PROG
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(Python)
from sympy import divisor_sigma
from itertools import accumulate
def f(_, n): return _ + 2*divisor_sigma(n, 1)
def aupton(terms): return list(accumulate(range(terms+1), f))[1:]
(PARI) a(n) = 2*sum(k=1, n, sigma(k)); \\ Michel Marcus, Dec 20 2021
(Python)
from math import isqrt
def A327329(n): return -(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1, s+1)) # Chai Wah Wu, Oct 22 2023
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CROSSREFS
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Cf. A001105, A006218, A013661, A024916, A067436, A222548, A236104, A237591, A237593, A243980, A245092, A262626.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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