A091912 -id:A091912 - cp's OEIS frontend

A241601 Largest divisor of A246006(n) whose prime factors are all >= n+2.

1, 1, 1, 1, 1, 1, 1, 61, 1, 277, 1, 50521, 691, 41581, 1, 199360981, 3617, 228135437, 43867, 2404879675441, 174611, 14814847529501, 77683, 69348874393137901, 236364091, 238685140977801337, 657931, 4087072509293123892361, 3392780147, 454540704683713199807

Offset: 0

Eric Chen, Dec 15 2014

nonn

Notice: Not all a(n) are 1 or primes, the first example is a(11) = 50521, it equals 19*2659.
a(2n) is a product of powers of Bernoulli irregular primes (A000928), with the exception of n = 0,1,2,3,4,5,7.
a(2n+1) is a product of powers of Euler irregular primes (A120337), with the exception of n = 0,1,2.
Conjectures: All terms are squarefree, and there are infinitely many n such that a(n) is prime.
a(n) = 1 iff n is in the set {0, 1, 2, 3, 4, 5, 6, 8, 10, 14}.
a(n) is prime for n = {7, 9, 12, 16, 17, 18, 26, 34, 36, 38, 39, 42, 49, 74, 114, 118, ...}.
All prime factors of a(n) are irregular primes (Bernoulli or Euler) and with an irregular pair to n: (61, 7), (277, 9), (19, 11), (2659, 11), (691, 12), (43, 13), (967, 13), (47, 15), (4241723, 15), (3617, 16), (228135437, 17), (43867, 18), (79, 19), (349, 19), (84224971, 19), ...
Number of ns such that a prime p divides a(n) is the irregular index of p, for example, 67 divides both a(27) and a(58), so it has irregular index two.
a(149) is the first a(n) which is not completely factored (with a 202-digit composite remaining).

Cf. A246006.
Cf. A000928, A120337.
Cf. A001067, A120082, A141590, A060054, A091912.

b[n_] := Numerator[BernoulliB[2 n]/(2 n)];
c[n_] := Numerator[SeriesCoefficient[Log[Tan[x]+1/Cos[x]], {x, 0, 2n+1}]];
a[0] = 1; a[n_] := If[EvenQ[n], b[n/2] // Abs, c[(n-1)/2]];
Table[a[n], {n, 0, 29}] (* Jean-François Alcover, Jul 03 2019 *)

a(2n) = |A001067(n)| = |A120082(2n)| = |A141590(n)| = |A060054(n)|.

a(2n+1) = A091912(n).

A013523 Denominator of [x^(2n+1)] in the Taylor expansion arcsinh(cosec(x) - cot(x)).

Original entry on oeis.org

2, 48, 768, 645120, 37158912, 81749606400, 784796221440, 42849873690624000, 548478383239987200, 63777066403145711616000, 4285818862291391820595200, 216862434431944426122117120000, 8007228348256409579893555200000, 1461479318123759876522171695104000000

Offset: 0

Patrick Demichel (patrick.demichel(AT)hp.com)

Numerators are apparently covered by A091912.

The e.g.f. of x/2, arcsinh(cosec(x)-cot(x)) = x/(2^1*1!) + x^3/(2^3*3!) + 5*x^5/(2^5*5!) + 61*x^7/(2^7*7!) + 1385*x^9/(2^9*9!) + ... is apparently provided by the absolute values of A028296.

			x/2 + x^3/48 + x^5/768 + 61*x^7/645120 + 277*x^9/37158912 + ...

G. C. Greubel, Table of n, a(n) for n = 0..200

Denominator[Take[CoefficientList[Series[ArcSinh[Csc[x] - Cot[x]], {x,0,25}], x], {2, -1, 2}]] (* G. C. Greubel, Nov 12 2016 *)

Name edited by R. J. Mathar, Dec 19 2011

A136606 Reduced denominators in the Maclaurin series for the Gudermannian.

Original entry on oeis.org

1, 6, 24, 5040, 72576, 39916800, 95800320, 1307674368000, 4184557977600, 121645100408832000, 2043637686868377600, 25852016738884976640000, 238634000666630553600000, 10888869450418352160768000000

Offset: 1

Eric W. Weisstein, Jan 12 2008

nonn

			x - x^3/6 + x^5/24 - (61*x^7)/5040 + (277*x^9)/72576 - ...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 33, equation 33:14:14 at page 318.

Eric Weisstein's World of Mathematics, Gudermannian.
Eric Weisstein's World of Mathematics, Inverse Gudermannian.

Cf. A091912.

gd[x_] := 2*ArcTan[(E^x - 1)/(E^x + 1)]; CoefficientList[ Series[gd[x], {x, 0, 28}]/x , x^2] // Denominator (* Jean-François Alcover, Nov 09 2012 *)

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A241601 Largest divisor of A246006(n) whose prime factors are all >= n+2.

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A013523 Denominator of [x^(2n+1)] in the Taylor expansion arcsinh(cosec(x) - cot(x)).

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A136606 Reduced denominators in the Maclaurin series for the Gudermannian.

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