A283533
a(n) = Sum_{d|n} d^(2*d + 1).
Original entry on oeis.org
1, 33, 2188, 262177, 48828126, 13060696236, 4747561509944, 2251799813947425, 1350851717672994277, 1000000000000048828158, 895430243255237372246532, 953962166440690142662256812, 1192533292512492016559195008118
Offset: 1
a(6) = 1^(2+1) + 2^(4+1) + 3^(6+1) + 6^(12+1) = 13060696236.
Cf. Sum_{d|n} d^(k*d+1):
A283498 (k=1), this sequence (k=2),
A283535 (k=3).
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f[n_] := Block[{d = Divisors[n]}, Total[d^(2 d + 1)]]; Array[f, 14] (* Robert G. Wilson v, Mar 10 2017 *)
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a(n) = sumdiv(n, d, d^(2*d+1)); \\ Michel Marcus, Mar 11 2017
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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))))) \\ Seiichi Manyama, Jun 18 2019
A308698
Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*d).
Original entry on oeis.org
1, 1, 2, 1, 5, 2, 1, 17, 28, 3, 1, 65, 730, 261, 2, 1, 257, 19684, 65553, 3126, 4, 1, 1025, 531442, 16777281, 9765626, 46688, 2, 1, 4097, 14348908, 4294967553, 30517578126, 2176783082, 823544, 4, 1, 16385, 387420490, 1099511628801, 95367431640626, 101559956688164, 678223072850, 16777477, 3
Offset: 1
Square array begins:
1, 1, 1, 1, 1, ...
2, 5, 17, 65, 257, ...
2, 28, 730, 19684, 531442, ...
3, 261, 65553, 16777281, 4294967553, ...
2, 3126, 9765626, 30517578126, 95367431640626, ...
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T[n_, k_] := DivisorSum[n, #^(k*#) &]; Table[T[k, n - k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, May 09 2021 *)
A308753
a(n) = Sum_{d|n} d^(2*(d-1)).
Original entry on oeis.org
1, 5, 82, 4101, 390626, 60466262, 13841287202, 4398046515205, 1853020188851923, 1000000000000390630, 672749994932560009202, 552061438912436478063702, 542800770374370512771595362, 629983141281877223617054459942
Offset: 1
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a[n_] := DivisorSum[n, #^(2*(# - 1)) &]; Array[a, 14] (* Amiram Eldar, May 08 2021 *)
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{a(n) = sumdiv(n, d, d^(2*(d-1)))}
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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-3)))))
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N=20; x='x+O('x^N); Vec(sum(k=1, N, k^(2*(k-1))*x^k/(1-x^k)))
A308756
a(n) = Sum_{d|n} d^(2*(d-2)).
Original entry on oeis.org
1, 2, 10, 258, 15626, 1679627, 282475250, 68719476994, 22876792454971, 10000000000015627, 5559917313492231482, 3833759992447476802059, 3211838877954855105157370, 3214199700417740937033562867, 3787675244106352329254150406260
Offset: 1
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a[n_] := DivisorSum[n, #^(2*(# - 2)) &]; Array[a, 15] (* Amiram Eldar, May 08 2021 *)
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{a(n) = sumdiv(n, d, d^(2*(d-2)))}
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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-5)))))
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N=20; x='x+O('x^N); Vec(sum(k=1, N, k^(2*(k-2))*x^k/(1-x^k)))
A359732
a(n) = Sum_{d|n} d^(2*d-1).
Original entry on oeis.org
1, 9, 244, 16393, 1953126, 362797308, 96889010408, 35184372105225, 16677181699666813, 10000000000001953134, 7400249944258160101212, 6624737266949237373933820, 7056410014866816666030739694, 8819763977946281130541873428720
Offset: 1
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a[n_] := DivisorSum[n, #^(2*# - 1) &]; Array[a, 15] (* Amiram Eldar, Aug 14 2023 *)
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a(n) = sumdiv(n, d, d^(2*d-1));
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my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, k^(2*k-1)*x^k/(1-x^k)))
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