A350146 -id:A350146 - cp's OEIS frontend

A350143 a(n) = Sum_{k=1..n} floor(n/(2*k-1))^2.

1, 4, 10, 17, 27, 41, 55, 70, 93, 115, 137, 167, 193, 223, 267, 298, 332, 381, 419, 465, 525, 571, 617, 679, 738, 792, 868, 930, 988, 1080, 1142, 1205, 1297, 1367, 1459, 1560, 1634, 1712, 1820, 1914, 1996, 2120, 2206, 2300, 2450, 2544, 2638, 2764, 2875, 2996, 3136, 3246

Offset: 1

Seiichi Manyama, Dec 16 2021

nonn

Winston de Greef, Table of n, a(n) for n = 1..10000

Column 2 of A350122.

Cf. A001227, A002131, A060831, A350146.

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

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

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

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

a(n) = Sum_{k=1..n} Sum_{d|k, k/d odd} 2*d - 1 = Sum_{k=1..n} 2 * A002131(k) - A001227(k) = 2 * A350146(n) - A060831(n).

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

A350166 Partial sums of A050469.

Original entry on oeis.org

1, 3, 5, 9, 15, 19, 25, 33, 40, 52, 62, 70, 84, 96, 108, 124, 142, 156, 174, 198, 210, 230, 252, 268, 299, 327, 347, 371, 401, 425, 455, 487, 507, 543, 579, 607, 645, 681, 709, 757, 799, 823, 865, 905, 947, 991, 1037, 1069, 1112, 1174, 1210, 1266, 1320, 1360, 1420

Offset: 1

Seiichi Manyama, Dec 18 2021

nonn

Winston de Greef, Table of n, a(n) for n = 1..10000

Cf. A050469, A101455, A350146.

f[2, e_] := 2^e; f[p_, e_] := If[Mod[p, 4] == 1, (p^(e + 1) - 1)/(p - 1), (p^(e + 1) + (-1)^e)/(p + 1)]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate @ Array[s, 50] (* Amiram Eldar, Dec 18 2021 *)

a(n) = sum(k=1, n, sumdiv(k, d, kronecker(-4, k/d)*d));

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

a(n) = Sum_{k=1..n} Sum_{d|k} A101455(k/d) * d = Sum_{k=1..n} A050469(k).

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

cp's OEIS Frontend

A350143 a(n) = Sum_{k=1..n} floor(n/(2*k-1))^2.

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A350166 Partial sums of A050469.

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Author

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Programs

Mathematica

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Formula