A350147 -id:A350147 - cp's OEIS frontend

A350145 a(n) = Sum_{k=1..n} floor(n/(2*k-1))^n.

1, 4, 28, 257, 3127, 46721, 823673, 16777474, 387440175, 10000060075, 285311849809, 8916117229571, 302875173709313, 11112007094026243, 437893920912819179, 18446744226340554502, 827240262649405488542, 39346408176856882188621

Offset: 1

Seiichi Manyama, Dec 16 2021

nonn

Main diagonal of A350122.

Cf. A350147.

a[n_] := Sum[Floor[n/(2*k - 1)]^n, {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Dec 17 2021 *)

PARI
```
a(n) = sum(k=1, n, (n\(2*k-1))^n);
```

a(n) = sum(k=1, n, sumdiv(k, d, k/d%2*(d^n-(d-1)^n)));

a(n) = Sum_{k=1..n} Sum_{d|k, k/d odd} d^n - (d - 1)^n.
a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} (k^n - (k - 1)^n) * x^k/(1 - x^(2*k)).
a(n) ~ n^n. - Vaclav Kotesovec, Dec 17 2021

A350167 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/(2*k-1))^k.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 3, 4, 1, 9, 9, 3, 5, -9, 1, 2, 4, 25, 25, 63, -13, -75, -75, -89, -26, 102, 296, 122, 124, -58, -58, -57, -741, -229, -471, 288, 290, -732, 1302, 1472, 1474, 2824, 2824, -542, -4556, -8650, -8650, -8680, -9783, -1320, 17818, 32016, 32018, 20252, 9054, 7360

Offset: 1

Seiichi Manyama, Dec 18 2021

sign

Winston de Greef, Table of n, a(n) for n = 1..1500

Cf. A350147, A350164.

a[n_] := Sum[(-1)^(k + 1) * Floor[n/(2*k - 1)]^k, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Dec 18 2021 *)

a(n) = sum(k=1, n, (-1)^(k+1)*(n\(2*k-1))^k);

my(N=66, x='x+O('x^N)); Vec(-sum(j=1, N, (1-x^(2*j-1))*sum(k=1, N, (-k)^j*x^(k*(2*j-1))))/(1-x))

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

A350238 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor((n/(2*k-1))^k).

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 3, 4, 4, 4, 3, 4, 6, 4, 3, 4, 6, 4, 3, 6, 5, 5, 6, 5, 7, 4, 4, 4, 5, 3, 6, 5, 5, 6, 5, 5, 4, 6, 5, 5, 8, 7, 6, 4, 7, 6, 3, 5, 6, 6, 5, 5, 8, 5, 5, 4, 4, 6, 5, 3, 6, 4, 7, 6, 8, 6, 5, 5, 7, 6, 4, 4, 7, 4, 5, 9, 8, 5, 6, 6, 8, 7, 3, 6, 6, 7, 7, 4, 9, 10, 3, 8, 4, 7, 6, 8, 10, 6, 4, 11, 7, 7, 5, 7, 11, 6, 8, 8, 9

Offset: 1

Seiichi Manyama, Dec 21 2021

sign

a(895) = -5.

			a(3) = [3/1] - [(3/3)^2] = 3 - 1 = 2.
a(4) = [4/1] - [(4/3)^2] = 4 - 1 = 3.
a(5) = [5/1] - [(5/3)^2] + [(5/5)^3] = 5 - 2 + 1 = 4.

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A350147, A350167, A350223, A350239.

a[n_] := Sum[(-1)^(k+1) * Floor[(n/(2*k-1))^k], {k, 1, n}]; Array[a, 100] (* Amiram Eldar, Dec 21 2021 *)

a(n) = sum(k=1, (n+1)\2, (-1)^(k+1)*(n^k\(2*k-1)^k));

cp's OEIS Frontend

A350145 a(n) = Sum_{k=1..n} floor(n/(2*k-1))^n.

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A350167 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/(2*k-1))^k.

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A350238 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor((n/(2*k-1))^k).

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