A308594 -id:A308594 - cp's OEIS frontend

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

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.

A308623 Number of ways to write n as a(a+1)/2 + b(b+1)/2 + 2^c*10^(2d) with a,b,c,d nonnegative integers.

Original entry on oeis.org

1, 2, 2, 3, 3, 2, 3, 5, 2, 4, 4, 3, 3, 5, 3, 3, 6, 4, 4, 5, 2, 6, 5, 4, 4, 5, 2, 4, 6, 3, 4, 8, 5, 3, 5, 4, 5, 8, 5, 5, 4, 3, 5, 7, 4, 5, 8, 4, 2, 8, 2, 6, 7, 4, 3, 4, 6, 5, 8, 4, 4, 6, 5, 5, 5, 5, 6, 8, 4, 6, 7, 4, 6, 10, 4, 4, 7, 5, 2, 10, 4, 7, 7, 4, 8, 4, 4, 7, 8, 2, 4, 9, 5, 5, 9, 5, 5, 7, 5, 6

Offset: 1

Zhi-Wei Sun, Jun 11 2019

nonn

Conjecture: a(n) > 0 for all n > 0. Equivalently, any positive integer n can be written as a*(a+1)/2 + b*(b+1)/2 + 2^c*10^(2d) with a,b,c,d nonnegative integers.
This was motivated by A308566, and we verified a(n) > 0 for all n = 1..2*10^8. Then, on the author's request, Giovanni Resta verified the above conjecture for n up to 10^10. G. Resta also noted that 729546026 cannot be written as a*(a+1)/2 + b*(b+1)/2 + 2^c*3^d with a,b,c,d nonnegative integers.
See also A308566, A308594 and A308621 for similar conjectures.

			a(1) = 1 with 1 = 0*1/2 + 0*1/2 + 2^0*10^(2*0).
a(10107) = 1 with 10107 = 82*83/2 + 96*97/2 + 2^11*10^(2*0).

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000079, A000217, A011557, A308566, A308584, A308621.

TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
tab={};Do[r=0;Do[If[TQ[n-10^(2k)*2^m-x(x+1)/2],r=r+1],{k,0,Log[10,n]/2},{m,0,Log[2,n/10^(2k)]},{x,0,(Sqrt[4(n-10^(2k)*2^m)+1]-1)/2}];tab=Append[tab,r],{n,1,100}];Print[tab]

A308668 a(n) = Sum_{d|n} d^(n/d+n).

Original entry on oeis.org

1, 9, 82, 1089, 15626, 287010, 5764802, 135270401, 3487315843, 100244173394, 3138428376722, 107072686593858, 3937376385699290, 155601328490478978, 6568412173896940652, 295165920677390712833, 14063084452067724991010

Offset: 1

Seiichi Manyama, Jun 16 2019

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

Diagonal of A308502.

Cf. A152211, A294956, A308594.

a[n_] := DivisorSum[n, #^(n/# + n) &]; Array[a, 20] (* Amiram Eldar, Mar 17 2021 *)

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

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

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, k^(k+1)*x^k/(1-k^(k+1)*x^k))) \\ Seiichi Manyama, Mar 17 2021

from sympy import divisors
def A308668(n): return sum(d**(n//d+n) for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 19 2022

L.g.f.: -log(Product_{k>=1} (1 - k*(k*x)^k)^(1/k)) = Sum_{k>=1} a(k)*x^k/k.
G.f.: Sum_{k>=1} k^(k+1) * x^k/(1 - k^(k+1) * x^k). - Seiichi Manyama, Mar 17 2021
a(n) ~ n^(n+1). - Vaclav Kotesovec, Aug 30 2025

A359882 a(n) = Sum_{d|n} d^n * (n/d)^d.

Original entry on oeis.org

1, 6, 30, 324, 3130, 53070, 823550, 17829896, 387951939, 10312525610, 285311670622, 9056807631948, 302875106592266, 11198819379685518, 437901307945957140, 18518802767263301648, 827240261886336764194, 39423330565860716459946

Offset: 1

Seiichi Manyama, Jan 16 2023

nonn

Cf. A066108, A308594, A359103.

a[n_] := DivisorSum[n, #^n * (n/#)^# &]; Array[a, 20] (* Amiram Eldar, Aug 09 2023 *)

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

a(n) = [x^n] Sum_{k>0} (n * x)^k / (1 - (k * x)^k).

If p is prime, a(p) = p + p^p.

cp's OEIS Frontend

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

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A308623 Number of ways to write n as a*(a+1)/2 + b*(b+1)/2 + 2^c*10^(2d) with a,b,c,d nonnegative integers.

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Mathematica

A308668 a(n) = Sum_{d|n} d^(n/d+n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Python

Formula

A359882 a(n) = Sum_{d|n} d^n * (n/d)^d.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A308623 Number of ways to write n as a(a+1)/2 + b(b+1)/2 + 2^c*10^(2d) with a,b,c,d nonnegative integers.