A050469 -id:A050469 - cp's OEIS frontend

A364031 Expansion of Sum_{k>0} k * x^k / (1 + x^(4*k)).

1, 2, 3, 4, 4, 6, 7, 8, 10, 8, 11, 12, 12, 14, 12, 16, 18, 20, 19, 16, 20, 22, 23, 24, 21, 24, 30, 28, 28, 24, 31, 32, 34, 36, 28, 40, 36, 38, 36, 32, 42, 40, 43, 44, 40, 46, 47, 48, 50, 42, 54, 48, 52, 60, 44, 56, 58, 56, 59, 48, 60, 62, 67, 64, 48, 68, 67, 72, 68, 56, 71, 80, 74, 72, 63, 76, 76, 72, 79

Offset: 1

Seiichi Manyama, Jul 01 2023

nonn

Cf. A050469, A364012, A364019.

Cf. A363037.

a[n_] := DivisorSum[n, (-1)^((# - 1)/4) * n/# &, Mod[#, 4] == 1 &]; Array[a, 100] (* Amiram Eldar, Jul 03 2023 *)

a(n) = sumdiv(n, d, (d%4==1)*(-1)^((d-1)/4)*n/d);

G.f.: Sum_{k>0} (-1)^(k-1) * x^(4*k-3) / (1 - x^(4*k-3))^2.

a(n) = Sum_{d|n, d==1 (mod 4)} (-1)^((d-1)/4) * (n/d).

A326575 Expansion of Sum_{k>=1} k * x^k * (1 + x^(2k)) / (1 + x^(2k) + x^(4*k)).

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 8, 8, 9, 8, 10, 12, 14, 16, 12, 16, 16, 18, 20, 16, 24, 20, 22, 24, 21, 28, 27, 32, 28, 24, 32, 32, 30, 32, 32, 36, 38, 40, 42, 32, 40, 48, 44, 40, 36, 44, 46, 48, 57, 42, 48, 56, 52, 54, 40, 64, 60, 56, 58, 48, 62, 64, 72, 64, 56, 60

Offset: 1

Ilya Gutkovskiy, Sep 12 2019

			G.f. = x + 2*x^2 + 3*x^3 + 4*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 8*x^8 + ... - _Michael Somos_, Oct 23 2019

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A003586 (fixed points), A035178, A050469, A122373, A326401.

Cf. A175646, A340578.

nmax = 66; CoefficientList[Series[Sum[k x^k (1 + x^(2 k))/(1 + x^(2 k) + x^(4 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, # &, MemberQ[{1}, Mod[n/#, 6]] &] - DivisorSum[n, # &, MemberQ[{5}, Mod[n/#, 6]] &], {n, 1, 66}]
f[p_, e_] := Which[p < 5, p^e, Mod[p, 6] == 5, (p^(e + 1) - (-1)^(e + 1))/(p + 1), Mod[p, 6] == 1, (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)

a(n) = { sumdiv(n, d, d*((n/d%6==1)-(n/d%6==5))) } \\ Andrew Howroyd, Sep 12 2019

{a(n) = if( n<0, 0, sumdiv( n, d, n/d * kronecker( -12, d)))}; /* Michael Somos, Oct 23 2019 */

a(n) = Sum_{d|n, n/d==1 (mod 6)} d - Sum_{d|n, n/d==5 (mod 6)} d.
G.f.: Sum_{k>=0}  x^(6*k+1) / (1 - x^(6*k+1))^2 - x^(6*k+5) / (1 - x^(6*k+5))^2. - Michael Somos, Oct 23 2019
Multiplicative with a(p^e) = p^e if p < 5, (p^(e+1)-(-1)^(e+1))/(p+1) if p == 5 (mod 6), and (p^(e+1)-1)/(p-1) if p == 1 (mod 6). - Amiram Eldar, Dec 02 2020
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{primes p == 5 (mod 6)} 1/(1+1/p^2) * Product_{primes p == 1 (mod 3)} 1/(1 - 1/p^2) = A340578 * A175646 / 2 = 0.48831400806... . - Amiram Eldar, Nov 06 2022

A330511 Expansion of e.g.f. Sum_{k>=1} arctan(x^k).

Original entry on oeis.org

1, 2, 4, 24, 144, 480, 4320, 40320, 282240, 4354560, 36288000, 319334400, 6706022400, 74724249600, 1046139494400, 20922789888000, 376610217984000, 4979623993344000, 115242726703104000, 2919482409811968000, 29194824098119680000

Offset: 1

Ilya Gutkovskiy, Dec 16 2019

nonn

Cf. A005359, A050469, A176475, A330504, A330505.

nmax = 21; CoefficientList[Series[Sum[ArcTan[x^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[(n - 1)! DivisorSum[n, (-1)^((n/# - 1)/2) # &, OddQ[n/#] &], {n, 1, 21}]

a(n) = (n-1)!*sumdiv(n, d, if (n/d % 2, (-1)^((n/d - 1)/2)*d)); \\ Michel Marcus, Dec 17 2019

E.g.f.: Sum_{i>=1} Sum_{j>=1} (-1)^(j + 1) * x^(i*(2*j - 1)) / (2*j - 1).

a(n) = (n - 1)! * Sum_{d|n, n/d odd} (-1)^((n/d - 1)/2) * d.

cp's OEIS Frontend

A364031 Expansion of Sum_{k>0} k * x^k / (1 + x^(4*k)).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A326575 Expansion of Sum_{k>=1} k * x^k * (1 + x^(2k)) / (1 + x^(2k) + x^(4*k)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A330511 Expansion of e.g.f. Sum_{k>=1} arctan(x^k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

cp's OEIS Frontend

A364031 Expansion of Sum_{k>0} k * x^k / (1 + x^(4*k)).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A326575 Expansion of Sum_{k>=1} k * x^k * (1 + x^(2*k)) / (1 + x^(2*k) + x^(4*k)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A330511 Expansion of e.g.f. Sum_{k>=1} arctan(x^k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A326575 Expansion of Sum_{k>=1} k * x^k * (1 + x^(2k)) / (1 + x^(2k) + x^(4*k)).