A345176 -id:A345176 - cp's OEIS frontend

A347415 a(n) = Sum_{k=1..n} floor((n/k)^k).

1, 3, 6, 11, 18, 31, 48, 76, 118, 184, 279, 426, 641, 966, 1448, 2163, 3228, 4805, 7137, 10586, 15681, 23198, 34278, 50606, 74632, 109987, 161954, 238312, 350432, 514999, 756407, 1110391, 1629219, 2389346, 3502578, 5132354, 7517523, 11007078, 16110784, 23573102, 34480937, 50420909

Offset: 1

Seiichi Manyama, Aug 31 2021

nonn

			a(3) = [3/1] + [(3/2)^2] + [(3/3)^3] = 3 + 2 + 1 = 6.

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

Cf. A006218, A062071, A153818, A344675, A345176, A347416.

a[n_] := Sum[Floor[(n/k)^k], {k, 1, n}]; Array[a, 40] (* Amiram Eldar, Aug 31 2021 *)

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

a(n) ~ sqrt(2*Pi*n) * exp(exp(-1)*n - 1/2). - Vaclav Kotesovec, Sep 14 2021

A350147 a(n) = Sum_{k=1..n} floor(n/(2*k-1))^k.

Original entry on oeis.org

1, 2, 4, 5, 7, 11, 13, 14, 21, 29, 31, 39, 41, 57, 87, 88, 90, 133, 135, 173, 253, 317, 319, 335, 398, 526, 756, 932, 934, 1300, 1302, 1303, 1991, 2503, 3001, 3806, 3808, 4832, 6918, 7088, 7090, 9836, 9838, 13206, 21860, 25956, 25958, 25990, 27097, 35560, 54766

Offset: 1

Seiichi Manyama, Dec 16 2021

nonn

Cf. A345176, A350145.

a[n_] := Sum[Floor[n/(2*k - 1)]^k, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Dec 17 2021 *)

PARI
```
a(n) = sum(k=1, n, (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)).

Limit_{n->infinity} a(n)^(1/n) = exp(exp(-1)/2). - Vaclav Kotesovec, Dec 17 2021

cp's OEIS Frontend

A347415 a(n) = Sum_{k=1..n} floor((n/k)^k).

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A350147 a(n) = Sum_{k=1..n} floor(n/(2*k-1))^k.

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