A283498 -id:A283498 - cp's OEIS frontend

A283499 Expansion of exp( Sum_{n>=1} -A283498(n)/n*x^n ) in powers of x.

1, -1, -4, -23, -223, -2767, -42268, -759008, -15672223, -365639304, -9512549191, -273072804420, -8575012101043, -292422232720311, -10762617713743350, -425245537127322111, -17953822507629389009, -806668679245000383731

Offset: 0

Seiichi Manyama, Mar 09 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..386

Cf. A023880, A283498.

A[n_] :=  Sum[d^(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, 17}] (* Indranil Ghosh, Mar 11 2017 *)

a(n) = if(n==0, 1, -(1/n)*sum(k=1, n, sumdiv(k, d, d^(d + 1))*a(n - k)));
for(n=0, 20, print1(a(n), ", ")) \\ Indranil Ghosh, Mar 11 2017

G.f.: Product_{k>=1} (1 - x^k)^(k^k).

a(n) = -(1/n)*Sum_{k=1..n} A283498(k)*a(n-k) for n > 0.

A023880 Number of partitions in expanding space.

Original entry on oeis.org

1, 1, 5, 32, 298, 3531, 51609, 894834, 17980052, 410817517, 10518031721, 298207687029, 9273094072138, 313757506696967, 11474218056441581, 450961669608632160, 18954582520550896213, 848384721904740036422, 40285256621556957160307, 2022695276960566890383148

Offset: 0

Olivier Gérard

nonn

Also partitions of n into 1 sort of 1, 4 sorts of 2, 27 sorts of 3, ..., k^k sorts of k. - Joerg Arndt, Feb 04 2015

Alois P. Heinz, Table of n, a(n) for n = 0..300

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^(k^k): k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(
      add(d*d^d, d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 04 2015

nmax=20; CoefficientList[Series[Product[1/(1-x^k)^(k^k),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 14 2015 *)

m=30; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^(k^k))) \\ G. C. Greubel, Oct 31 2018

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: n^n)
print([b(n) for n in range(20)]) # Peter Luschny, Nov 11 2020

G.f.: 1 / Product_{k>=1} (1 - x^k)^(k^k).
a(n) ~ n^n * (1 + exp(-1)/n + (exp(-1)/2 + 5*exp(-2))/n^2). - Vaclav Kotesovec, Mar 14 2015
a(n) = (1/n)*Sum_{k=1..n} A283498(k)*a(n-k) for n > 0. - Seiichi Manyama, 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

Seiichi Manyama, Mar 10 2017

nonn

Inverse Mobius transform of A085526. - R. J. Mathar, Mar 11 2017

			a(6) = 1^(2+1) + 2^(4+1) + 3^(6+1) + 6^(12+1) = 13060696236.

Seiichi Manyama, Table of n, a(n) for n = 1..214

Cf. Sum_{d|n} d^(k*d+1): A283498 (k=1), this sequence (k=2), A283535 (k=3).

Cf. A308696.

f[n_] := Block[{d = Divisors[n]}, Total[d^(2 d + 1)]]; Array[f, 14] (* Robert G. Wilson v, Mar 10 2017 *)

a(n) = sumdiv(n, d, d^(2*d+1)); \\ Michel Marcus, Mar 11 2017

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

L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^(2*k))) = Sum_{k>=1} a(k)*x^k/k. - Seiichi Manyama, Jun 18 2019

A283535 a(n) = Sum_{d|n} d^(3*d + 1).

Original entry on oeis.org

1, 129, 59050, 67108993, 152587890626, 609359740069674, 3909821048582988050, 37778931862957228818561, 523347633027360537213570571, 10000000000000000000152587890754, 255476698618765889551019445759400442, 8505622499821102144576132293474637113130

Offset: 1

Seiichi Manyama, Mar 10 2017

nonn

			a(6) = 1^(3+1) + 2^(6+1) + 3^(9+1) + 6^(18+1) = 609359740069674.

Seiichi Manyama, Table of n, a(n) for n = 1..152

Cf. Sum_{d|n} d^(k*d+1): A283498 (k=1), A283533 (k=2), this sequence (k=3).

Cf. A308697.

f[n_] := Block[{d = Divisors[n]}, Total[d^(3 d + 1)]]; Array[f, 12] (* Robert G. Wilson v, Mar 10 2017 *)

a(n) = sumdiv(n, d, d^(3*d+1)); \\ Michel Marcus, Mar 11 2017

N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-x^k)^k^(3*k))))) \\ Seiichi Manyama, Jun 18 2019

L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^(3*k))) = Sum_{k>=1} a(k)*x^k/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

Seiichi Manyama, Mar 17 2017

nonn

			a(6) = 1^(4+1) + 2^(8+1) + 3^(12+1) + 6^(24+1) = 28430288029931296212.

Seiichi Manyama, Table of n, a(n) for n = 1..119

Cf. Sum_{d|n} d^(k*d+1): A283498 (k=1), A283533 (k=2), A283535 (k=3), this sequence (k=4).

Table[Sum[d^(4*d + 1), {d, Divisors[n]}], {n, 20}] (* Indranil Ghosh, Mar 17 2017 *)

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

Seiichi Manyama, Jun 18 2019

			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, ...

Seiichi Manyama, Antidiagonals n = 1..52, flattened

Columns k=0..3 give A000203, A283498, A283533, A283535.
Row n=1..2 give A000012, A087289.
Cf. A294579, A308698, A308701.

T[n_, k_] := DivisorSum[n, #^(k*# + 1) &]; Table[T[k, n - k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, May 09 2021 *)

L.g.f. of column k: -log(Product_{j>=1} (1 - x^j)^(j^(k*j))).

G.f. of column k: Sum_{j>=1} j^(k*j+1) * x^j/(1 - x^j).

A308755 a(n) = Sum_{d|n} d^(d-2).

Original entry on oeis.org

1, 2, 4, 18, 126, 1301, 16808, 262162, 4782973, 100000127, 2357947692, 61917365541, 1792160394038, 56693912392105, 1946195068359504, 72057594038190098, 2862423051509815794, 121439531096599036046, 5480386857784802185940, 262144000000000100000143

Offset: 1

Seiichi Manyama, Jun 22 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..388

Cf. A062796, A262843, A283498, A308753.

a[n_] := DivisorSum[n, #^(# - 2) &]; Array[a, 20] (* Amiram Eldar, May 08 2021 *)

PARI
```
{a(n) = sumdiv(n, d, d^(d-2))}
```

N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-x^k)^k^(k-3)))))

N=20; x='x+O('x^N); Vec(sum(k=1, N, k^(k-2)*x^k/(1-x^k)))

L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^(k-3))) = Sum_{k>=1} a(k)*x^k/k.

G.f.: Sum_{k>=1} k^(k-2) * x^k/(1 - x^k).

A304870 L.g.f.: log(Product_{k>=1} (1 + x^k)^(k^k)) = Sum_{n>=1} a(n)*x^n/n.

Original entry on oeis.org

1, 7, 82, 1015, 15626, 279862, 5764802, 134216695, 3486784483, 99999984382, 3138428376722, 106993205100070, 3937376385699290, 155568095552047430, 6568408355712906332, 295147905179218607095, 14063084452067724991010, 708235345355334189853093, 37589973457545958193355602

Offset: 1

Ilya Gutkovskiy, May 20 2018

nonn

			L.g.f.: L(x) = x + 7*x^2/2 + 82*x^3/3 + 1015*x^4/4 + 15626*x^5/5 + 279862*x^6/6 + 5764802*x^7/7 + 134216695*x^8/8 + 3486784483*x^9/9 + ...
exp(L(x)) = 1 + x + 4*x^2 + 31*x^3 + 289*x^4 + 3495*x^5 + 51268*x^6 + 891152*x^7 + 17926913*x^8 + 409907600*x^9 + ... + A261053(n)*x^n + ...

Index entries for sequences related to sums of divisors

Cf. A007778, A062796, A261053, A283498.

nmax = 19; Rest[CoefficientList[Series[Log[Product[(1 + x^k)^k^k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 19; Rest[CoefficientList[Series[Sum[k^(k + 1) x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
a[n_] := Sum[(-1)^(n/d + 1) d^(d + 1), {d, Divisors[n]}]; Table[a[n], {n, 1, 19}]

G.f.: Sum_{k>=1} k^(k+1)*x^k/(1 + x^k).
a(n) = Sum_{d|n} (-1)^(n/d+1)*d^(d+1).
a(p) = p^(p+1) + 1 where p is an odd prime.

cp's OEIS Frontend

A283499 Expansion of exp( Sum_{n>=1} -A283498(n)/n*x^n ) in powers of x.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A023880 Number of partitions in expanding space.

Views

Author

Keywords

Comments

Links

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

A283533 a(n) = Sum_{d|n} d^(2*d + 1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A283535 a(n) = Sum_{d|n} d^(3*d + 1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A283369 a(n) = Sum_{d|n} d^(4*d + 1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A308755 a(n) = Sum_{d|n} d^(d-2).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI