A062796 -id:A062796 - cp's OEIS frontend

A283335 Expansion of exp( Sum_{n>=1} -A062796(n)/n*x^n ) in powers of x.

1, -1, -2, -7, -54, -544, -7005, -108220, -1958263, -40629205, -951376217, -24826365255, -714568797261, -22491957589783, -768651303338761, -28344950796904518, -1121910285249842486, -47442295013058570884, -2134673855370621621400

Offset: 0

Seiichi Manyama, Mar 08 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..387

Cf. A023879 (exp( Sum_{n>=1} A062796(n)/n*x^n )), A062796.

A[n_] :=  Sum[d^d, {d, Divisors[n]}]; a[n_] := If[n==0, 1, -(1/n)*Sum[A[k]*a[n - k], {k, n}]]; Table[a[n], {n, 0, 18}] (* Indranil Ghosh, Mar 11 2017 *)

a(n) = if(n==0, 1, -(1/n)*sum(k=1, n, sumdiv(k, d, d^d)*a(n - k)));
for(n=0, 18, print1(a(n), ", ")) \\ Indranil Ghosh, Mar 11 2017

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

a(n) = -(1/n)*Sum_{k=1..n} A062796(k)*a(n-k) for n > 0.

A342628 a(n) = Sum_{d|n} d^(n-d).

Original entry on oeis.org

1, 2, 2, 6, 2, 45, 2, 322, 731, 3383, 2, 132901, 2, 827641, 10297068, 33570818, 2, 2578617270, 2, 44812807567, 678610493340, 285312719189, 2, 393061010002613, 95367431640627, 302875123369471, 150094917726535604, 569939345952661545, 2, 105474306078445349841, 2

Offset: 1

Seiichi Manyama, Mar 16 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..599

Cf. A023887, A062796, A308594, A342629.

a[n_] := DivisorSum[n, #^(n - #) &]; Array[a, 30] (* Amiram Eldar, Mar 17 2021 *)

PARI
```
a(n) = sumdiv(n, d, d^(n-d));
```

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-(k*x)^k)))

from sympy import divisors
def A342628(n): return sum(d**(n-d) for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 19 2022

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

If p is prime, a(p) = 2.

A262843 Inverse Moebius transform of n^(n-1).

Original entry on oeis.org

1, 3, 10, 67, 626, 7788, 117650, 2097219, 43046731, 1000000628, 25937424602, 743008378540, 23298085122482, 793714773371796, 29192926025391260, 1152921504608944195, 48661191875666868482, 2185911559738739586477, 104127350297911241532842, 5242880000000001000000692, 278218429446951548637314060

Offset: 1

Paul D. Hanna, Oct 03 2015

nonn

Logarithmic derivative of A262842.

			O.g.f.: A(x) = x + 3*x^2 + 10*x^3 + 67*x^4 + 626*x^5 + 7788*x^6 +...
where
A(x) = x/(1-x) + 2*x^2/(1-x^2) + 3^2*x^3/(1-x^3) + 4^3*x^4/(1-x^4) + 5^4*x^5/(1-x^5) + 6^5*x^6/(1-x^6) +...+ n^(n-1)* x^n/(1 -x^n) +...
Logarithmic generating function.
L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 67*x^4/4 + 626*x^5/5 + 7788*x^6/6 +...
where
exp(L(x)) = 1/( (1-x) * (1-x^2) * (1-x^3)^3 * (1-x^4)^16 * (1-x^5)^125 * (1-x^6)^1296 *...* (1-x^n)^(n^(n-2)) *...).
Explicitly,
exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 22*x^4 + 150*x^5 + 1469*x^6 + 18452*x^7 + 282426*x^8 +...+ A262842(n)*x^n ...

G. C. Greubel, Table of n, a(n) for n = 1..385

Cf. A262842, A062796.

a[n_] := DivisorSum[n, #^(#-1) &]; Array[a, 30] (* Jean-François Alcover, Dec 23 2015 *)

{a(n)=sumdiv(n,d, d^(d-1))}
for(n=1,30,print1(a(n),", "))

{a(n)=polcoeff(sum(m=1, n, m^(m-1)*x^m/(1-x^m +x*O(x^n))), n)}
for(n=1,30,print1(a(n),", "))

from sympy import divisors
def A262843(n): return sum(d**(d-1) for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 19 2022

a(n) = Sum{d|n} d^(d-1).

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

A066108 Sum n^d over all divisors of n.

Original entry on oeis.org

1, 6, 30, 276, 3130, 46914, 823550, 16781384, 387421227, 10000100110, 285311670622, 8916103456860, 302875106592266, 11112006930971730, 437893890381622140, 18446744078004584720, 827240261886336764194, 39346408075494930884190, 1978419655660313589123998

Offset: 1

Labos Elemer, Dec 05 2001

nonn

This is neither the Moebius transform nor the inverse Moebius transform of n^n, although it is close to them.

			n = 12: a(12) = A066106(12) = 8916103456860 = 8916100448256+2985984+20736+1728+144+12.
For comparison: M-transform of n^n at 12 = 8916100401348 = 8916100448256-46656-256+0+4+0 = A062793(12);
Inverse M-transform of n^n at 12 = 8916100495200 = 8916100448256+46656+256+27+4+1 = A062796(12).

Harry J. Smith, Table of n, a(n) for n = 1..100

Cf. A062793, A062796.

A066108 := n -> add(n^d, d = NumberTheory[Divisors](n)); seq(A066108(n), n = 1 .. 19) - Miles Wilson, Jan 12 2025

Table[Sum[n^d, {d, Divisors@ n}], {n, 19}] (* Michael De Vlieger, Dec 20 2015 *)

a(n)=sumdiv(n,d, n^d );  /* Joerg Arndt, Oct 07 2012 */

a(n) = Sum_{d|n} n^d.
a(n) ~ n^n. - Vaclav Kotesovec, Jun 05 2021
Conjectured g.f.: Sum_{m>=1} (m^m*Sum_{k=1..m}(x^(k*m)*Sum_{j=0..k}((-1)^j*(k - j)^m*binomial(m + 1, j)))/(1 - x^m)^(m + 1)), where the inner summation is the Triangle of Eulerian numbers A008292. - Miles Wilson, Jan 12 2025

A283498 a(n) = Sum_{d|n} d^(d+1).

Original entry on oeis.org

1, 9, 82, 1033, 15626, 280026, 5764802, 134218761, 3486784483, 100000015634, 3138428376722, 106993205660122, 3937376385699290, 155568095563577034, 6568408355712906332, 295147905179487044617, 14063084452067724991010, 708235345355341163422059, 37589973457545958193355602

Offset: 1

Seiichi Manyama, Mar 09 2017

nonn

			a(6) = 1^2 + 2^3 + 3^4 + 6^7 = 280026.

Seiichi Manyama, Table of n, a(n) for n = 1..385

Cf. A007778, A062796 (Sum_{d|n} d^d).

f[n_] := Block[{d = Divisors[n]}, Total[d^(d + 1)]]; Array[f, 19] (* Robert G. Wilson v, Mar 10 2017 *)

a(n) = sumdiv(n, d, d^(d+1)); \\ Michel Marcus, Mar 09 2017

from sympy import divisors
def A283498(n): return sum(d**(d+1) for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 19 2022

From Ilya Gutkovskiy, May 06 2017: (Start)
G.f.: Sum_{k>=1} k^(k+1)*x^k/(1 - x^k).
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^k)) = Sum_{n>=1} a(n)*x^n/n. (End)

More terms from Michel Marcus, Mar 09 2017

A294956 a(n) = Sum_{d|n} d^(d + n/d).

Original entry on oeis.org

1, 9, 82, 1041, 15626, 280212, 5764802, 134221889, 3486785131, 100000078254, 3138428376722, 106993207077516, 3937376385699290, 155568095598166344, 6568408355713287812, 295147905180426634241, 14063084452067724991010

Offset: 1

Seiichi Manyama, Nov 12 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..385

Cf. A055225, A062796, A294957.

sd[n_]:=Total[#^(#+n/#)&/@Divisors[n]]; Array[sd,20] (* Harvey P. Dale, Mar 28 2021 *)

PARI
```
a(n) = sumdiv(n, d, d^(d+n/d));
```

N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-k*x^k)^(k^(k-1)))))) \\ Seiichi Manyama, Jun 09 2019

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, k^(k+1)*x^k/(1-k*x^k))) \\ Seiichi Manyama, Jan 11 2023

L.g.f.: -log(Product_{k>=1} (1 - k*x^k)^(k^(k-1))) = Sum_{k>=1} a(k)*x^k/k. - Seiichi Manyama, Jun 09 2019

G.f.: Sum_{k>0} k^(k+1) * x^k / (1 - k * x^k). - Seiichi Manyama, Jan 11 2023

A023879 Number of partitions in expanding space.

Original entry on oeis.org

1, 1, 3, 12, 79, 722, 8675, 128177, 2248873, 45644104, 1051632553, 27107038863, 772751427746, 24136897360750, 819689757351091, 30068876227952332, 1184869328943005936, 49914047187427191742

Offset: 0

Olivier Gérard

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..380

Cf. A062796.

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

seq(coeff(series(mul((1-x^k)^(-k^(k-1)),k=1..n),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Oct 31 2018

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

{a(n)=polcoeff(prod(k=1,n,(1-x^k+x*O(x^n))^(-k^(k-1))),n)}

{a(n)=polcoeff(exp(sum(m=1,n+1,sumdiv(m, d, d^d)*x^m/m) +x*O(x^n)),n)} \\ Paul D. Hanna, Sep 05 2012

G.f.: Product_{k>=1} (1 - x^k)^(-k^(k-1)).
G.f.: exp( Sum_{n>=1} A062796(n)/n*x^n ), where A062796(n) = Sum_{d|n} d^d. - Paul D. Hanna, Sep 05 2012
a(n) ~ n^(n-1). - Vaclav Kotesovec, Mar 14 2015

A174472 a(n) = Sum_{d|n} d^sigma(d).

Original entry on oeis.org

1, 9, 82, 16393, 15626, 2176782426, 5764802, 35184372105225, 2541865828411, 1000000000000015634, 3138428376722, 1648446623609512543953220489306, 3937376385699290, 3214199700417740936756852426, 16834112196028232574462906332, 21267647932558653966460948148857618441

Offset: 1

Paul D. Hanna, Apr 04 2010

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..167

Cf. A000203 (sigma), A062796, A174471, A174937, A345895.

a[n_] := DivisorSum[n, #^DivisorSigma[1, #] &]; Array[a, 16] (* Amiram Eldar, Oct 08 2021 *)

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

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, k^sigma(k)*x^k/(1-x^k))) \\ Seiichi Manyama, Oct 14 2021

Logarithmic derivative of A174471.

G.f.: Sum_{k>=1} k^sigma(k) * x^k/(1 - x^k). - Seiichi Manyama, Oct 14 2021

A308594 a(n) = Sum_{d|n} d^(d+n).

Original entry on oeis.org

1, 17, 730, 65601, 9765626, 2176802276, 678223072850, 281474993488897, 150094635297530563, 100000000030517582222, 81402749386839761113322, 79496847203492408399442540, 91733330193268616658399616010, 123476695691248494372093865205800

Offset: 1

Seiichi Manyama, Jun 09 2019

nonn

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

Cf. A062796, A294956, A308668.

sp[n_]:=Module[{d=Divisors[n]},Table[d[[k]]^(d[[k]]+n),{k,Length[ d]}]] // Total; Array[sp,15] (* Harvey P. Dale, Jan 02 2020 *)
a[n_] := DivisorSum[n, #^(# + n) &]; Array[a, 14] (* Amiram Eldar, May 11 2021 *)

PARI
```
a(n) = sumdiv(n, d, d^(d+n));
```

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

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k^2*x)^k/(1-(k*x)^k))) \\ Seiichi Manyama, Mar 16 2021

from sympy import divisors
def A308594(n): return sum(d**(d+n) for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 19 2022

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

G.f.: Sum_{k>=1} (k^2 * x)^k/(1 - (k * x)^k). - Seiichi Manyama, Mar 16 2021

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

Seiichi Manyama, Jun 17 2019

			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, ...

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

Columns k=0..3 give A000005, A062796, A308696, A308697.
Row n=1..2 give A000012, A052539.
Cf. A308509, A308701, A308704.

T[n_, k_] := DivisorSum[n, #^(k*#) &]; 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-1))).

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

cp's OEIS Frontend

A283335 Expansion of exp( Sum_{n>=1} -A062796(n)/n*x^n ) in powers of x.

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A342628 a(n) = Sum_{d|n} d^(n-d).

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A262843 Inverse Moebius transform of n^(n-1).

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Mathematica

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A066108 Sum n^d over all divisors of n.

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A283498 a(n) = Sum_{d|n} d^(d+1).

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A294956 a(n) = Sum_{d|n} d^(d + n/d).

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A023879 Number of partitions in expanding space.

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