A309126 -id:A309126 - cp's OEIS frontend

A309125 a(n) = n + 2^2 * floor(n/2^2) + 3^2 * floor(n/3^2) + 4^2 * floor(n/4^2) + ...

1, 2, 3, 8, 9, 10, 11, 16, 26, 27, 28, 33, 34, 35, 36, 57, 58, 68, 69, 74, 75, 76, 77, 82, 108, 109, 119, 124, 125, 126, 127, 148, 149, 150, 151, 201, 202, 203, 204, 209, 210, 211, 212, 217, 227, 228, 229, 250, 300, 326, 327, 332, 333, 343, 344, 349, 350, 351, 352, 357, 358, 359, 369, 454, 455, 456

Offset: 1

Ilya Gutkovskiy, Jul 13 2019

nonn

Partial sums of A035316.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of a(n)/n^(3/2) for n = 1..10000

Cf. A013936, A024916, A035316, A309126, A309127.

Table[Sum[k^2 Floor[n/k^2], {k, 1, n}], {n, 1, 66}]
nmax = 66; CoefficientList[Series[1/(1 - x) Sum[k^2 x^(k^2)/(1 - x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}], {x, 0, nmax}], x] // Rest

a(n) = sum(k=1, n, k^2*(n\k^2)); \\ Seiichi Manyama, Aug 30 2021

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

a(n) ~ zeta(3/2)*n^(3/2)/3 - n/2. - Vaclav Kotesovec, Aug 30 2021

A309127 a(n) = n + 2^4 * floor(n/2^4) + 3^4 * floor(n/3^4) + 4^4 * floor(n/4^4) + ...

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139

Offset: 1

Ilya Gutkovskiy, Jul 13 2019

nonn

Partial sums of A300909.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A013938, A024916, A300909, A309125, A309126.

Table[Sum[k^4 Floor[n/k^4], {k, 1, n}], {n, 1, 75}]
nmax = 75; CoefficientList[Series[1/(1 - x) Sum[k^4 x^(k^4)/(1 - x^(k^4)), {k, 1, Floor[nmax^(1/4)] + 1}], {x, 0, nmax}], x] // Rest

a(n) = sum(k=1, n, k^4*(n\k^4)); \\ Seiichi Manyama, Aug 30 2021

G.f.: (1/(1 - x)) * Sum_{k>=1} k^4 * x^(k^4)/(1 - x^(k^4)).

a(n) ~ zeta(5/4)*n^(5/4)/5 - n/2. - Vaclav Kotesovec, Aug 30 2021

A347397 a(n) = Sum_{k=1..n} k^k * floor(n/k^k).

Original entry on oeis.org

1, 2, 3, 8, 9, 10, 11, 16, 17, 18, 19, 24, 25, 26, 27, 32, 33, 34, 35, 40, 41, 42, 43, 48, 49, 50, 78, 83, 84, 85, 86, 91, 92, 93, 94, 99, 100, 101, 102, 107, 108, 109, 110, 115, 116, 117, 118, 123, 124, 125, 126, 131, 132, 160, 161, 166, 167, 168, 169, 174, 175, 176, 177, 182, 183, 184

Offset: 1

Seiichi Manyama, Aug 30 2021

nonn

What is the limit_{n->infinity} a(n) / (n*log(n)/LambertW(log(n))) ?. - Vaclav Kotesovec, Aug 30 2021

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of a(n) / (n*log(n)/LambertW(log(n))) for n = 1..10000

Cf. A024916, A062071, A309125, A309126, A309127.

Table[Sum[k^k*Floor[n/k^k], {k, 1, n}], {n, 1, 100}] (* Vaclav Kotesovec, Aug 30 2021 *)

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

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

cp's OEIS Frontend

A309125 a(n) = n + 2^2 * floor(n/2^2) + 3^2 * floor(n/3^2) + 4^2 * floor(n/4^2) + ...

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A309127 a(n) = n + 2^4 * floor(n/2^4) + 3^4 * floor(n/3^4) + 4^4 * floor(n/4^4) + ...

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A347397 a(n) = Sum_{k=1..n} k^k * floor(n/k^k).

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Mathematica

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