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A356131
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a(n) = Sum_{k=1..n} (k - 1)^n * binomial(floor(n/k)+1,2).
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3
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0, 1, 9, 100, 1302, 20648, 377022, 7921039, 186926431, 4916562309, 142373072781, 4506381442625, 154721361953489, 5729251983077521, 227585590018322461, 9654855432715969784, 435659531345223039702, 20836069677785611552293
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OFFSET
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1,3
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LINKS
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FORMULA
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a(n) = Sum_{k=1..n} k * (sigma_{n-1}(k) - floor(n/k)^n) = A356129(n) - A350109(n).
a(n) = Sum_{k=1..n} k * Sum_{d|k} (d - 1)^n / d.
a(n) = [x^n] (1/(1-x)) * Sum_{k>=1} (k - 1)^n * x^k/(1 - x^k)^2.
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MATHEMATICA
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a[n_] := Sum[(k - 1)^n * Binomial[Floor[n/k]+1, 2], {k, 1, n}]; Array[a, 18] (* Amiram Eldar, Jul 28 2022 *)
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PROG
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(PARI) a(n) = sum(k=1, n, (k-1)^n*binomial((n\k)+1, 2));
(PARI) a(n) = sum(k=1, n, k*(sigma(k, n-1)-(n\k)^n));
(PARI) a(n) = sum(k=1, n, k*sumdiv(k, d, (d-1)^n/d));
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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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