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A326826
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a(n) = (1/2) * Sum_{d|n} (sigma_1(d) + sigma_2(d)), where sigma_1 = A000203 and sigma_2 = A001157.
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1
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1, 5, 8, 19, 17, 43, 30, 69, 60, 95, 68, 176, 93, 171, 166, 255, 155, 342, 192, 403, 303, 395, 278, 681, 358, 543, 490, 738, 437, 961, 498, 969, 709, 911, 720, 1476, 705, 1131, 978, 1603, 863, 1773, 948, 1732, 1440, 1643, 1130, 2634, 1284, 2110, 1648, 2391, 1433, 2882, 1706
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OFFSET
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1,2
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COMMENTS
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Inverse Moebius transform applied twice to triangular numbers (A000217).
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LINKS
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FORMULA
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G.f.: Sum_{i>=1} Sum_{j>=1} x^(i*j) / (1 - x^(i*j))^3.
G.f.: (1/2) * Sum_{i>=1} Sum_{j>=1} j * (j + 1) * x^(i*j) / (1 - x^(i*j)).
G.f.: (1/2) * Sum_{k>=1} (sigma_1(k) + sigma_2(k)) * x^k / (1 - x^k).
Dirichlet g.f.: zeta(s)^2 * (zeta(s-1) + zeta(s-2)) / 2.
a(n) = (1/2) * Sum_{d|n} d * (d + 1) * tau(n/d), where tau = A000005.
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MAPLE
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with(numtheory):
a:= n-> add(d*(d+1)*tau(n/d), d=divisors(n))/2:
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MATHEMATICA
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Table[1/2 Sum[DivisorSigma[1, d] + DivisorSigma[2, d], {d, Divisors[n]}], {n, 1, 55}]
Table[1/2 Sum[d (d + 1) DivisorSigma[0, n/d], {d, Divisors[n]}], {n, 1, 55}]
nmax = 55; CoefficientList[Series[Sum[Sum[x^(i j)/(1 - x^(i j))^3, {j, 1, nmax}], {i, 1, nmax}], {x, 0, nmax}], x] // Rest
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PROG
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(PARI) a(n) = sumdiv(n, d, sigma(d)+sigma(d, 2))/2; \\ Michel Marcus, Oct 20 2019
(Magma) [(1/2)*&+[DivisorSigma(1, d)+DivisorSigma(2, d):d in Divisors(n)]:n in [1..55]]; // Marius A. Burtea, Oct 20 2019
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CROSSREFS
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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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