A344726
Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} (-1)^(j+1) * floor(n/j)^k.
Original entry on oeis.org
1, 1, 1, 1, 3, 3, 1, 7, 9, 2, 1, 15, 27, 12, 4, 1, 31, 81, 56, 22, 4, 1, 63, 243, 240, 118, 30, 6, 1, 127, 729, 992, 610, 196, 44, 4, 1, 255, 2187, 4032, 3094, 1230, 324, 48, 7, 1, 511, 6561, 16256, 15562, 7564, 2336, 448, 71, 7, 1, 1023, 19683, 65280, 77998, 45990, 16596, 3840, 685, 83, 9
Offset: 1
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 3, 7, 15, 31, 63, ...
3, 9, 27, 81, 243, 729, ...
2, 12, 56, 240, 992, 4032, ...
4, 22, 118, 610, 3094, 15562, ...
4, 30, 196, 1230, 7564, 45990, ...
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T[n_, k_] := Sum[(-1)^(j + 1) * Quotient[n, j]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, May 27 2021 *)
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T(n, k) = sum(j=1, n, (-1)^(j+1)*(n\j)^k);
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T(n, k) = sum(j=1, n, sumdiv(j, d, (-1)^(j/d+1)*(d^k-(d-1)^k)));
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));
A350122
Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} floor(n/(2*j-1))^k.
Original entry on oeis.org
1, 1, 2, 1, 4, 4, 1, 8, 10, 5, 1, 16, 28, 17, 7, 1, 32, 82, 65, 27, 9, 1, 64, 244, 257, 127, 41, 11, 1, 128, 730, 1025, 627, 225, 55, 12, 1, 256, 2188, 4097, 3127, 1313, 353, 70, 15, 1, 512, 6562, 16385, 15627, 7809, 2419, 522, 93, 17, 1, 1024, 19684, 65537, 78127, 46721, 16841, 4114, 759, 115, 19
Offset: 1
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
2, 4, 8, 16, 32, 64, 128, ...
4, 10, 28, 82, 244, 730, 2188, ...
5, 17, 65, 257, 1025, 4097, 16385, ...
7, 27, 127, 627, 3127, 15627, 78127, ...
9, 41, 225, 1313, 7809, 46721, 280065, ...
11, 55, 353, 2419, 16841, 117715, 823673, ...
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T[n_, k_] := Sum[Floor[n/(2*j - 1)]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Dec 17 2021 *)
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T(n, k) = sum(j=1, n, (n\(2*j-1))^k);
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T(n, k) = sum(j=1, n, sumdiv(j, d, j/d%2*(d^k-(d-1)^k)));
A356250
Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} (j * floor(n/j))^k.
Original entry on oeis.org
1, 1, 2, 1, 4, 3, 1, 8, 8, 4, 1, 16, 22, 15, 5, 1, 32, 62, 57, 21, 6, 1, 64, 178, 219, 91, 33, 7, 1, 128, 518, 849, 405, 185, 41, 8, 1, 256, 1522, 3315, 1843, 1053, 247, 56, 9, 1, 512, 4502, 13017, 8541, 6065, 1523, 402, 69, 10, 1, 1024, 13378, 51339, 40171, 35253, 9571, 2948, 545, 87, 11
Offset: 1
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
2, 4, 8, 16, 32, 64, 128, ...
3, 8, 22, 62, 178, 518, 1522, ...
4, 15, 57, 219, 849, 3315, 13017, ...
5, 21, 91, 405, 1843, 8541, 40171, ...
6, 33, 185, 1053, 6065, 35253, 206345, ...
7, 41, 247, 1523, 9571, 61091, 394987, ...
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T[n_, k_] := Sum[(j * Floor[n/j])^k, {j, 1, n}]; Table[T[k, n - k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Jul 31 2022 *)
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T(n, k) = sum(j=1, n, (j*(n\j))^k);
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T(n, k) = if(k==0, n, sum(j=1, n, j^k*sumdiv(j, d, 1-(1-1/d)^k)));
Showing 1-4 of 4 results.