A090739 -id:A090739 - cp's OEIS frontend

A189806 Denominators of coefficients in the series expansion of ((2 - m) EllipticK(m) - 2 EllipticE(m))/(Pi * m).

1, 16, 64, 2048, 8192, 262144, 1048576, 67108864, 268435456, 17179869184, 68719476736, 2199023255552, 8796093022208, 281474976710656, 1125899906842624, 144115188075855872, 576460752303423488, 73786976294838206464, 295147905179352825856, 9444732965739290427392, 37778931862957161709568

Offset: 0

Dan T. Abell, Apr 28 2011

This combination of elliptic functions appears in the expression for the vector potential generated by a circular loop of current. The denominators are powers of 2. The base-2 logarithm of the denominators increments in pattern related to A090739. That latter sequence begins 3,4,3,5,3,4,3,6. Add 2 to each entry; thus, 5,6,5,7,5,6,5,8. Duplicate each entry; thus, 5,5,6,6,5,5,7,7,5,5,6,6,5,5,8,8. Now insert a 2 at the beginning and between each entry; thus, 2,5,2,5,2,6,2,6,2,5,2,5,2, 7,2,7,2,5,2,5,2,6,2,6,2,5,2,5,2,8,2,8. Finally, prepend a 4; thus 4,2,5,2,5,2,6,2,6,2,5,2,5,2,7,2,7,2,5,2,5,2,6,2,6,2,5,2,5,2,8,2,8. This yields the pattern of increments in the base-2 logarithm of the denominators. See also the construction of the ruler sequence A007814.

J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, third edition, 1999, eq.(5.37).

G. C. Greubel, Table of n, a(n) for n = 0..830

Cf. A189805, A090739, A007814.

Denominator[CoefficientList[Series[((2-m)EllipticK[m]-2EllipticE[m])/m,{m,0,20}]/Pi,m]]

a(n) is the denominator of the fraction ((2n-1)!!)^2/(2^(2n+1)*(n-1)!*(n+1)!).

A091284 Exponent of 2 in -1+prime[n]^s, if s is an exponent of form 16k-8. Except a(1)=0, a(n)=1+A091283(n).

Original entry on oeis.org

0, 5, 5, 6, 5, 5, 7, 5, 6, 5, 8, 5, 6, 5, 7, 5, 5, 5, 5, 6, 6, 7, 5, 6, 8, 5, 6, 5, 5, 7, 10, 5, 6, 5, 5, 6, 5, 5, 6, 5, 5, 5, 9, 9, 5, 6, 5, 8, 5, 5

Offset: 1

Labos Elemer, Jan 22 2004

nonn

Exponents of 2 in -1+p^s if the exponent s[u]=(2^u)k-(2^(u-1) comes from other sequence generated with s[u-1] exponent by adding 1 to terms of the "previous" sequence. E.g. s=256k-128 needed an addition of 6 to the terms of A091282.

Cf. A023506, A007814, A091282, A091283, A090739, A090740, A090129.

Table[{8*k-4, Table[Part[Flatten[FactorInteger [ -1+Prime[n]^(16*k-8)]], 2], {n, 2, 50}]}, {k, 1, 2}]

A349693 Dirichlet convolution of the ruler function (A001511) with itself.

Original entry on oeis.org

1, 4, 2, 10, 2, 8, 2, 20, 3, 8, 2, 20, 2, 8, 4, 35, 2, 12, 2, 20, 4, 8, 2, 40, 3, 8, 4, 20, 2, 16, 2, 56, 4, 8, 4, 30, 2, 8, 4, 40, 2, 16, 2, 20, 6, 8, 2, 70, 3, 12, 4, 20, 2, 16, 4, 40, 4, 8, 2, 40, 2, 8, 6, 84, 4, 16, 2, 20, 4, 16, 2, 60, 2, 8, 6, 20, 4, 16, 2, 70

Offset: 1

Ilya Gutkovskiy, Nov 25 2021

Dirichlet convolution of A000005 with A104117. - Ridouane Oudra, Jul 23 2025

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000217, A001511, A115364, A129628.

Cf. A000005, A104117, A007814, A085058, A090739, A099777, A372784, A001227, A000292,

a:= n-> (f-> add(f(d)*f(n/d), d=numtheory[divisors](n)))(k-> padic[ordp](2*k, 2)):
seq(a(n), n=1..80);  # Alois P. Heinz, Nov 25 2021

Table[Sum[IntegerExponent[2 d, 2] IntegerExponent[2 n/d, 2], {d, Divisors[n]}], {n, 1, 80}]
f[p_, e_] := If[p == 2, Binomial[e + 3, 3], e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 80] (* Amiram Eldar, Nov 25 2021 *)

A001511(n) = (1+valuation(n,2));
A349693(n) = sumdiv(n,d,A001511(n/d)*A001511(d)); \\ Antti Karttunen, Nov 25 2021

from sympy import divisor_count
def A349693(n): return divisor_count(n)*(m:=(n&-n).bit_length()+1)*(m+1)//6 # Chai Wah Wu, Jul 13 2022

Dirichlet g.f.: zeta(s)^2 * 4^s / (2^s-1)^2.
a(n) = Sum_{d|n} A001511(d) * A001511(n/d).
a(n) = Sum_{d|n} A000217(A001511(d)).
Multiplicative with a(p^e) = binomial(e+3,3) if p = 2 and e+1 otherwise. - Amiram Eldar, Nov 25 2021
Sum_{k=1..n} a(k) ~ 4*n*(log(n) - 1 + 2*gamma - 2*log(2)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Nov 26 2021
From Ridouane Oudra, Jul 23 2025: (Start)
a(n) = Sum_{i=0..A007814(n)} (i+1)*tau(n/2^i).
a(n) = Sum_{d|n} A115364(d).
a(n) = (1/6)*A090739(n)*A085058(n-1)*A000005(n).
a(n) = (1/6)*A001511(n)*A090739(n)*A099777(n).
a(n) = (1/3)*A115364(n)*A372784(n).
a(n) = A001227(n)*A000292(A001511(n)).
a(2*n+1) = tau(2*n+1).
a(2^k*(2*n+1)) = binomial(k+3, 3)*tau(2*n+1), for k, n >= 0. (End)

A363228 Exponent of 4 in 9^n - 1.

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2

Offset: 1

Ruud H.G. van Tol, May 21 2023

Not the same as A147648-without-zeros.

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

Cf. A007814, A090739, A147648, A244415.

Cf. A024101, A235127.

a[n_] := IntegerExponent[2*n, 4] + 1; Array[a, 100] (* Amiram Eldar, May 22 2023 *)

PARI
```
a(n) = valuation(2*n, 4) + 1;
```

def A363228(n): return (~n&n-1).bit_length()+3>>1 # Chai Wah Wu, Jul 09 2023

a(n) = floor(A090739(n)/2).
a(n) = A244415(n) + 1.
a(n) = A235127(A024101(n)). - Michel Marcus, May 21 2023
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 5/3. - Amiram Eldar, Jul 13 2023
Conjecture: a(n) = A235127(A000045(6*n)), all other 4-adic 6-sections A235127(A000045(.))=0. - R. J. Mathar, Jun 28 2025

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A189806 Denominators of coefficients in the series expansion of ((2 - m) EllipticK(m) - 2 EllipticE(m))/(Pi * m).

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A091284 Exponent of 2 in -1+prime[n]^s, if s is an exponent of form 16k-8. Except a(1)=0, a(n)=1+A091283(n).

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A349693 Dirichlet convolution of the ruler function (A001511) with itself.

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A363228 Exponent of 4 in 9^n - 1.

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