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A320940
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a(n) = Sum_{d|n} d*sigma_n(d).
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4
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1, 11, 85, 1127, 15631, 287021, 5764809, 135007759, 3487020610, 100146496681, 3138428376733, 107032667155169, 3937376385699303, 155582338242604221, 6568408966322733475, 295154660699054931999, 14063084452067724991027, 708239400347943609329270
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = n * [x^n] -log(Product_{k>=1} (1 - x^k)^sigma_n(k)).
a(n) = Sum_{d|n} d^(n+1)*sigma_1(n/d).
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EXAMPLE
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a(6) = 1*sigma_6(1)+2*sigma_6(2)+3*sigma_6(3)+6*sigma_6(6) = 1+2*65+3*730+6*47450 = 287021.
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MAPLE
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with(numtheory): seq(coeff(series(n*(-log(mul((1-x^k)^sigma[n](k), k=1..n))), x, n+1), x, n), n = 1 .. 20); # Muniru A Asiru, Oct 28 2018
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MATHEMATICA
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Table[Sum[d DivisorSigma[n, d], {d, Divisors[n]}] , {n, 18}]
Table[n SeriesCoefficient[-Log[Product[(1 - x^k)^DivisorSigma[n, k], {k, 1, n}]], {x, 0, n}], {n, 18}]
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PROG
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(PARI) a(n) = sumdiv(n, d, d*sigma(d, n)); \\ Michel Marcus, Oct 28 2018
(Python)
from sympy import divisor_sigma, divisors
return sum(divisor_sigma(d)*(n//d)**(n+1) for d in divisors(n, generator=True)) # Chai Wah Wu, Feb 15 2020
(Magma) [&+[d*DivisorSigma(n, d):d in Divisors(n)]:n in [1..18]]; // Marius A. Burtea, Feb 15 2020
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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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