A308694 -id:A308694 - cp's OEIS frontend

A308692 a(n) = Sum_{d|n} d^(2*(n/d - 1)).

1, 2, 2, 6, 2, 27, 2, 82, 83, 283, 2, 2047, 2, 4147, 7188, 20546, 2, 125964, 2, 343407, 533844, 1048699, 2, 10076747, 390627, 16777387, 43053284, 84003927, 2, 667311413, 2, 1342439682, 3486799044, 4294967587, 249905428, 52916914768, 2, 68719477099, 282429565044

Offset: 1

Seiichi Manyama, Jun 17 2019

Robert Israel, Table of n, a(n) for n = 1..3143

Column k=2 of A308694.

N:=100: # for a(1)..a(N)
g:= add(x^k/(1-k^2*x^k),k=1..N):
S:= series(g,x,N+1):
seq(coeff(S,x,j),j=1..N); # Robert Israel, Apr 05 2020

a[n_] := DivisorSum[n, #^(2*(n/# - 1)) &]; Array[a, 39] (* Amiram Eldar, May 09 2021 *)

PARI
```
{a(n) = sumdiv(n, d, d^(2*(n/d-1)))}
```

N=66; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-k^2*x^k)^(1/k^3)))))

L.g.f.: -log(Product_{k>=1} (1 - k^2*x^k)^(1/k^3)) = Sum_{k>=1} a(k)*x^k/k.
a(p) = 2 for prime p.
G.f.: Sum_{k>=1} x^k/(1 - k^2*x^k). - Ilya Gutkovskiy, Jul 25 2019

A308690 Square array A(n,k), n >= 1, k >= 0, where A(n,k) = Sum_{d|n} d^(k*n/d - k + 1), read by antidiagonals.

Original entry on oeis.org

1, 1, 3, 1, 3, 4, 1, 3, 4, 7, 1, 3, 4, 9, 6, 1, 3, 4, 13, 6, 12, 1, 3, 4, 21, 6, 24, 8, 1, 3, 4, 37, 6, 66, 8, 15, 1, 3, 4, 69, 6, 216, 8, 41, 13, 1, 3, 4, 133, 6, 762, 8, 201, 37, 18, 1, 3, 4, 261, 6, 2784, 8, 1289, 253, 68, 12, 1, 3, 4, 517, 6, 10386, 8, 9225, 2197, 648, 12, 28

Offset: 1

Seiichi Manyama, Jun 17 2019

			Square array begins:
    1,  1,  1,   1,   1,    1,     1, ...
    3,  3,  3,   3,   3,    3,     3, ...
    4,  4,  4,   4,   4,    4,     4, ...
    7,  9, 13,  21,  37,   69,   133, ...
    6,  6,  6,   6,   6,    6,     6, ...
   12, 24, 66, 216, 762, 2784, 10386, ...
    8,  8,  8,   8,   8,    8,     8, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=0..3 give A000203, A055225, A308688, A308689.

Cf. A294579, A308509, A308694.

T[n_, k_] := DivisorSum[n, #^(k*n/# - k + 1) &]; Table[T[k, n - k], {n, 1, 12}, {k, 1, n}] // Flatten (* Amiram Eldar, May 09 2021 *)

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

A(p,k) = p+1 for prime p.

A308701 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, 2, 1, 3, 2, 1, 5, 10, 3, 1, 9, 82, 67, 2, 1, 17, 730, 4101, 626, 4, 1, 33, 6562, 262153, 390626, 7788, 2, 1, 65, 59050, 16777233, 244140626, 60466262, 117650, 4, 1, 129, 531442, 1073741857, 152587890626, 470184985314, 13841287202, 2097219, 3

Offset: 1

Seiichi Manyama, Jun 22 2019

			Square array begins:
   1,   1,      1,         1,            1, ...
   2,   3,      5,         9,           17, ...
   2,  10,     82,       730,         6562, ...
   3,  67,   4101,    262153,     16777233, ...
   2, 626, 390626, 244140626, 152587890626, ...

Seiichi Manyama, Antidiagonals n = 1..53, flattened

Columns k=0..2 give A000005, A262843, A308753.
Row n=1..3 give A000012, A000051, A062396.
Cf. A308694, A308698, A308704.

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-k-1))).

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

A308693 a(n) = Sum_{d|n} d^(3*(n/d - 1)).

Original entry on oeis.org

1, 2, 2, 10, 2, 93, 2, 578, 731, 4223, 2, 56765, 2, 262489, 547068, 2359810, 2, 31173510, 2, 152949071, 387538140, 1073743157, 2, 20134371189, 244140627, 68719478935, 282430067924, 618515646977, 2, 12056339359929, 2, 39582552821762, 205891133866212

Offset: 1

Seiichi Manyama, Jun 17 2019

nonn

Column k=3 of A308694.

a[n_] := DivisorSum[n, #^(3*(n/# - 1)) &]; Array[a, 33] (* Amiram Eldar, May 09 2021 *)

PARI
```
{a(n) = sumdiv(n, d, d^(3*(n/d-1)))}
```

N=66; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-k^3*x^k)^(1/k^4)))))

L.g.f.: -log(Product_{k>=1} (1 - k^3*x^k)^(1/k^4)) = Sum_{k>=1} a(k)*x^k/k.
a(p) = 2 for prime p.
G.f.: Sum_{k>=1} x^k/(1 - k^3*x^k). - Ilya Gutkovskiy, Jul 25 2019

cp's OEIS Frontend

A308692 a(n) = Sum_{d|n} d^(2*(n/d - 1)).

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A308690 Square array A(n,k), n >= 1, k >= 0, where A(n,k) = Sum_{d|n} d^(k*n/d - k + 1), read by antidiagonals.

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A308701 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*(d-1)).

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Examples

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Programs

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Formula

A308693 a(n) = Sum_{d|n} d^(3*(n/d - 1)).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

PARI

Formula