A300909 -id:A300909 - cp's OEIS frontend

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

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

A333844 G.f.: Sum_{k>=1} k * x^(k^4) / (1 - x^(k^4)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3

Offset: 1

Ilya Gutkovskiy, Apr 07 2020

Sum of 4th roots of 4th powers dividing n.

Amiram Eldar, Table of n, a(n) for n = 1..10000
A. Dixit, B. Maji, and A. Vatwani, Voronoi summation formula for the generalized divisor function sigma_z^k(n), arXiv:2303.09937 [math.NT], 2023, sigma(z=1,k=4,n).

Cf. A000203, A053164, A063775, A069290, A300909, A333843.

nmax = 112; CoefficientList[Series[Sum[k x^(k^4)/(1 - x^(k^4)), {k, 1, Floor[nmax^(1/4)] + 1}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, #^(1/4) &, IntegerQ[#^(1/4)] &], {n, 112}]
f[p_, e_] := (p^(Floor[e/4] + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 01 2020 *)

Dirichlet g.f.: zeta(s) * zeta(4*s-1).
If n = Product (p_j^k_j) then a(n) = Product ((p_j^(floor(k_j/4) + 1) - 1)/(p_j - 1)).
Sum_{k=1..n} a(k) ~ zeta(3)*n + zeta(1/2)*sqrt(n)/2. - Vaclav Kotesovec, Dec 01 2020

A347398 Expansion of g.f. Sum_{k>=1} k^k * x^(k^k)/(1 - x^(k^k)).

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 28, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 28, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 28, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 32, 1, 1, 1, 5

Offset: 1

Seiichi Manyama, Aug 30 2021

nonn

			1^1 | 108, 2^2 | 108 and 3^3 | 108. So a(108) = 1^1 + 2^2 + 3^3 = 32.

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

Cf. A000203, A000312, A035316, A113061, A300909, A347397, A347399.

PARI
```
a(n) = sum(k=1, n, (n%k^k==0)*k^k);
```

a(n) = A347397(n) - A347397(n-1) for n > 1.

a(n) = Sum_{k=1..n, k^k | n} k^k.

cp's OEIS Frontend

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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A333844 G.f.: Sum_{k>=1} k * x^(k^4) / (1 - x^(k^4)).

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A347398 Expansion of g.f. Sum_{k>=1} k^k * x^(k^k)/(1 - x^(k^k)).

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