A372619
Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = 1/(phi(k)) * Sum_{j=1..n} phi(k*j).
Original entry on oeis.org
1, 1, 2, 1, 3, 4, 1, 2, 5, 6, 1, 3, 5, 9, 10, 1, 2, 5, 7, 13, 12, 1, 3, 4, 9, 11, 17, 18, 1, 2, 6, 6, 13, 14, 23, 22, 1, 3, 4, 10, 11, 17, 20, 31, 28, 1, 2, 5, 6, 14, 13, 23, 24, 37, 32, 1, 3, 5, 9, 10, 20, 19, 31, 33, 45, 42, 1, 2, 5, 7, 13, 12, 26, 23, 37, 37, 55, 46
Offset: 1
Square array T(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
2, 3, 2, 3, 2, 3, 2, 3, 2, 3, ...
4, 5, 5, 5, 4, 6, 4, 5, 5, 5, ...
6, 9, 7, 9, 6, 10, 6, 9, 7, 9, ...
10, 13, 11, 13, 11, 14, 10, 13, 11, 14, ...
12, 17, 14, 17, 13, 20, 12, 17, 14, 18, ...
18, 23, 20, 23, 19, 26, 19, 23, 20, 24, ...
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T[n_, k_] := Sum[EulerPhi[k*j], {j, 1, n}] / EulerPhi[k]; Table[T[k, n-k+1], {n, 1, 12}, {k, 1, n}] // Flatten (* Amiram Eldar, May 09 2024 *)
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T(n, k) = sum(j=1, n, eulerphi(k*j))/eulerphi(k);
A372608
a(n) = Sum_{k=1..n} phi(n*k).
Original entry on oeis.org
1, 3, 10, 18, 44, 40, 114, 124, 198, 192, 430, 292, 708, 540, 704, 888, 1552, 954, 2178, 1456, 1980, 2080, 3806, 2216, 4220, 3480, 4734, 4056, 7588, 3560, 9270, 6960, 7920, 7840, 9936, 7296, 15588, 10980, 13056, 11120, 21240, 10128, 24570, 16360, 17880, 19360, 32062
Offset: 1
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Table[Sum[EulerPhi[n*k], {k, 1, n}], {n, 1, 50}] (* Vaclav Kotesovec, Aug 04 2025 *)
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a(n) = sum(k=1, n, eulerphi(k*n));
A372668
a(n) = (1/phi(n)) * Sum_{j=1..n} Sum_{k=1..n} phi(n*j*k).
Original entry on oeis.org
1, 9, 26, 83, 132, 404, 400, 989, 1199, 2382, 2096, 5381, 3922, 8358, 8525, 12897, 10758, 25517, 16618, 34116, 30217, 45156, 34224, 77503, 50559, 87512, 77328, 119162, 84364, 198907, 108928, 196605, 174258, 249884, 195499, 374490, 215930, 386822, 330878, 500717
Offset: 1
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Table[Sum[EulerPhi[n*j*k], {j, 1, n}, {k, 1, n}]/EulerPhi[n], {n, 1, 40}] (* Vaclav Kotesovec, May 10 2024 *)
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a(n) = sum(j=1, n, sum(k=1, n, eulerphi(n*j*k)))/eulerphi(n);
A372669
a(n) = Sum_{j=1..n} Sum_{k=1..n} phi(n*j*k) / phi(n*k).
Original entry on oeis.org
1, 6, 16, 37, 62, 121, 154, 256, 339, 499, 560, 884, 910, 1308, 1516, 1870, 1979, 2889, 2756, 3776, 4023, 4814, 4795, 6716, 6330, 7908, 8301, 9946, 9520, 13406, 11587, 14598, 15236, 17508, 17438, 22182, 19518, 24329, 24855, 29063, 26521, 35789, 30577, 37769
Offset: 1
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Table[Sum[EulerPhi[n*j*k] / EulerPhi[n*k], {j, 1, n}, {k, 1, n}], {n, 1, 50}] (* Vaclav Kotesovec, May 10 2024 *)
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a(n) = sum(j=1, n, sum(k=1, n, eulerphi(n*j*k)/eulerphi(n*k)));
Showing 1-4 of 4 results.
