A350125
a(n) = Sum_{k=1..n} k^2 * floor(n/k)^n.
Original entry on oeis.org
1, 8, 40, 345, 3303, 50225, 833569, 17045934, 388654659, 10039636255, 285508661853, 8924967326015, 302927979357701, 11114722212099135, 437913155876193839, 18447871416712820782, 827249276230172525622, 39347009369000530723017
Offset: 1
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a[n_] := Sum[k^2 * Floor[n/k]^n, {k, 1, n}]; Array[a, 18] (* Amiram Eldar, Oct 04 2023 *)
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a(n) = sum(k=1, n, k^2*(n\k)^n);
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a(n) = sum(k=1, n, k^2*sumdiv(k, d, (d^n-(d-1)^n)/d^2));
A350106
Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} j * floor(n/j)^k.
Original entry on oeis.org
1, 1, 4, 1, 6, 8, 1, 10, 14, 15, 1, 18, 32, 31, 21, 1, 34, 86, 87, 45, 33, 1, 66, 248, 295, 153, 81, 41, 1, 130, 734, 1095, 669, 309, 101, 56, 1, 258, 2192, 4231, 3201, 1521, 443, 150, 69, 1, 514, 6566, 16647, 15765, 8373, 2633, 722, 191, 87, 1, 1026, 19688, 66055, 78393, 48321, 17411, 4746, 1005, 253, 99
Offset: 1
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
4, 6, 10, 18, 34, 66, 130, ...
8, 14, 32, 86, 248, 734, 2192, ...
15, 31, 87, 295, 1095, 4231, 16647, ...
21, 45, 153, 669, 3201, 15765, 78393, ...
33, 81, 309, 1521, 8373, 48321, 284709, ...
41, 101, 443, 2633, 17411, 119321, 828323, ...
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T[n_, k_] := Sum[j * Floor[n/j]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Dec 14 2021 *)
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T(n, k) = sum(j=1, n, j*(n\j)^k);
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T(n, k) = sum(j=1, n, j*sumdiv(j, d, (d^k-(d-1)^k)/d));
A356131
a(n) = Sum_{k=1..n} (k - 1)^n * binomial(floor(n/k)+1,2).
Original entry on oeis.org
0, 1, 9, 100, 1302, 20648, 377022, 7921039, 186926431, 4916562309, 142373072781, 4506381442625, 154721361953489, 5729251983077521, 227585590018322461, 9654855432715969784, 435659531345223039702, 20836069677785611552293
Offset: 1
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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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a(n) = sum(k=1, n, (k-1)^n*binomial((n\k)+1, 2));
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a(n) = sum(k=1, n, k*(sigma(k, n-1)-(n\k)^n));
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a(n) = sum(k=1, n, k*sumdiv(k, d, (d-1)^n/d));
A356238
a(n) = Sum_{k=1..n} (k * floor(n/k))^n.
Original entry on oeis.org
1, 8, 62, 849, 8541, 206345, 2581403, 76623522, 1617299079, 49463993875, 952905453423, 59000021366675, 1198427462876421, 54128102218676115, 2321105129608323165, 117387839988330848902, 3205342976298888473968, 268263812478494295219717
Offset: 1
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a[n_] := Sum[(k * Floor[n/k])^n, {k, 1, n}]; Array[a, 18] (* Amiram Eldar, Jul 30 2022 *)
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a(n) = sum(k=1, n, (k*(n\k))^n);
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a(n) = sum(k=1, n, k^n*sumdiv(k, d, 1-(1-1/d)^n));
Showing 1-4 of 4 results.