A308696
a(n) = Sum_{d|n} d^(2*d).
Original entry on oeis.org
1, 17, 730, 65553, 9765626, 2176783082, 678223072850, 281474976776209, 150094635296999851, 100000000000009765642, 81402749386839761113322, 79496847203390846310290154, 91733330193268616658399616010, 123476695691247935826908004929122
Offset: 1
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a[n_] := DivisorSum[n, #^(2*#) &]; Array[a, 14] (* Amiram Eldar, May 09 2021 *)
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{a(n) = sumdiv(n, d, d^(2*d))}
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N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-x^k)^k^(2*k-1)))))
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N=20; x='x+O('x^N); Vec(sum(k=1, N, k^(2*k)*x^k/(1-x^k)))
A308697
a(n) = Sum_{d|n} d^(3*d).
Original entry on oeis.org
1, 65, 19684, 16777281, 30517578126, 101559956688164, 558545864083284008, 4722366482869661990977, 58149737003040059690409853, 1000000000000000000030517578190, 23225154419887808141001767796309132, 708801874985091845381344408569542626596
Offset: 1
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a[n_] := DivisorSum[n, #^(3*#) &]; Array[a, 12] (* Amiram Eldar, May 09 2021 *)
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{a(n) = sumdiv(n, d, d^(3*d))}
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N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-x^k)^k^(3*k-1)))))
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N=20; x='x+O('x^N); Vec(sum(k=1, N, k^(3*k)*x^k/(1-x^k)))
A308701
Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*(d-1)).
Original entry on oeis.org
1, 1, 2, 1, 3, 2, 1, 5, 10, 3, 1, 9, 82, 67, 2, 1, 17, 730, 4101, 626, 4, 1, 33, 6562, 262153, 390626, 7788, 2, 1, 65, 59050, 16777233, 244140626, 60466262, 117650, 4, 1, 129, 531442, 1073741857, 152587890626, 470184985314, 13841287202, 2097219, 3
Offset: 1
Square array begins:
1, 1, 1, 1, 1, ...
2, 3, 5, 9, 17, ...
2, 10, 82, 730, 6562, ...
3, 67, 4101, 262153, 16777233, ...
2, 626, 390626, 244140626, 152587890626, ...
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T[n_, k_] := DivisorSum[n, #^(k*(# - 1)) &]; Table[T[k, n - k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, May 09 2021 *)
A308704
Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*d+1).
Original entry on oeis.org
1, 1, 3, 1, 9, 4, 1, 33, 82, 7, 1, 129, 2188, 1033, 6, 1, 513, 59050, 262177, 15626, 12, 1, 2049, 1594324, 67108993, 48828126, 280026, 8, 1, 8193, 43046722, 17179869697, 152587890626, 13060696236, 5764802, 15, 1, 32769, 1162261468, 4398046513153, 476837158203126, 609359740069674, 4747561509944, 134218761, 13
Offset: 1
Square array begins:
1, 1, 1, 1, 1, ...
3, 9, 33, 129, 513, ...
4, 82, 2188, 59050, 1594324, ...
7, 1033, 262177, 67108993, 17179869697, ...
6, 15626, 48828126, 152587890626, 476837158203126, ...
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T[n_, k_] := DivisorSum[n, #^(k*# + 1) &]; Table[T[k, n - k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, May 09 2021 *)
Showing 1-4 of 4 results.