A283536
Expansion of exp( Sum_{n>=1} -A283535(n)/n*x^n ) in powers of x.
Original entry on oeis.org
1, -1, -64, -19619, -16755517, -30499543213, -101528172949440, -558442022082754554, -4721800698082895269442, -58144976385942395405449505, -999941534906642496357956893139, -23224150593200781968944997552887957, -708778584588517237886357058373629079824
Offset: 0
Cf. Product_{k>=1} (1 - x^k)^(k^(m*k)):
A010815 (m=0),
A283499 (m=1),
A283534 (m=2), this sequence (m=3).
Cf.
A283580 (Product_{k>=1} 1/(1 - x^k)^(k^(3*k))).
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A[n_] := Sum[d^(3*d + 1), {d, Divisors[n]}]; a[n_]:=If[n==0, 1, -(1/n)*Sum[A[k]*a[n - k], {k, n}]]; Table[a[n], {n, 0, 12}] (* Indranil Ghosh, Mar 11 2017 *)
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A(n) = sumdiv(n, d, d^(3*d + 1));
a(n) = if(n==0, 1, -(1/n)*sum(k=1, n, A(k)*a(n - k)));
for(n=0, 12, print1(a(n), ", ")) \\ Indranil Ghosh, Mar 11 2017
A283580
Expansion of exp( Sum_{n>=1} A283535(n)/n*x^n ) in powers of x.
Original entry on oeis.org
1, 1, 65, 19748, 16799044, 30535636881, 101591759812967, 558649739234980142, 4722932373908389412037, 58154498193439779564557624, 1000058469893323150011227885608, 23226158305362748824532880463596385, 708825166389400019044145225134521489486
Offset: 0
Cf. Product_{k>=1} 1/(1 - x^k)^(k^(m*k)):
A000041 (m=0),
A023880 (m=1),
A283579 (m=2), this sequence (m=3).
Cf.
A283536 (Product_{k>=1} (1 - x^k)^(k^(3*k))).
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A[n_] := Sum[d^(3*d + 1), {d, Divisors[n]}]; a[n_] := If[n==0, 1, (1/n)*Sum[A[k]*a[n - k], {k, n}]];Table[a[n], {n, 0, 12}] (* Indranil Ghosh, Mar 11 2017 *)
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A(n) = sumdiv(n, d, d^(3*d + 1));
a(n) = if(n==0, 1, (1/n)*sum(k=1, n, A(k)*a(n - k)));
for(n=0, 12, print1(a(n),", ")) \\ Indranil Ghosh, Mar 11 2017
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
A283369
a(n) = Sum_{d|n} d^(4*d + 1).
Original entry on oeis.org
1, 513, 1594324, 17179869697, 476837158203126, 28430288029931296212, 3219905755813179726837608, 633825300114114700765531472385, 202755595904452569706561330874548093, 100000000000000000000000000476837158203638
Offset: 1
a(6) = 1^(4+1) + 2^(8+1) + 3^(12+1) + 6^(24+1) = 28430288029931296212.
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Table[Sum[d^(4*d + 1), {d, Divisors[n]}], {n, 20}] (* Indranil Ghosh, Mar 17 2017 *)
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for(n=1, 20, print1(sumdiv(n, d, d^(4*d + 1)),", ")) \\ Indranil Ghosh, Mar 17 2017
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 *)
A308757
a(n) = Sum_{d|n} d^(3*(d-2)).
Original entry on oeis.org
1, 2, 28, 4098, 1953126, 2176782365, 4747561509944, 18014398509486082, 109418989131512359237, 1000000000000000001953127, 13109994191499930367061460372, 237376313799769806328952468217885, 5756130429098929077956071497934208654
Offset: 1
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a[n_] := DivisorSum[n, #^(3*(# - 2)) &]; Array[a, 13] (* Amiram Eldar, May 08 2021 *)
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{a(n) = sumdiv(n, d, d^(3*(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^(3*k-7)))))
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N=20; x='x+O('x^N); Vec(sum(k=1, N, k^(3*(k-2))*x^k/(1-x^k)))
Showing 1-6 of 6 results.
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