A006863 -id:A006863 - cp's OEIS frontend

A075180 Denominators from e.g.f. 1/(1-exp(-x)) - 1/x.

2, 12, 1, 120, 1, 252, 1, 240, 1, 132, 1, 32760, 1, 12, 1, 8160, 1, 14364, 1, 6600, 1, 276, 1, 65520, 1, 12, 1, 3480, 1, 85932, 1, 16320, 1, 12, 1, 69090840, 1, 12, 1, 541200, 1, 75852, 1, 2760, 1, 564, 1, 2227680, 1, 132, 1, 6360, 1, 43092, 1, 6960, 1, 708, 1, 3407203800, 1, 12, 1, 32640, 1, 388332, 1, 120, 1, 9372, 1, 10087262640, 1, 12

Offset: 0

Wolfdieter Lang, Sep 06 2002

Denominators of -zeta(-n), n >= 0, where zeta is Riemann's zeta function.

Numerators are +1, A060054(n+1), n >= 1.

			1/2, 1/12, 0, -1/120, 0, 1/252, 0, -1/240, 0, 1/132, 0, -691/32760, ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 807, combined eqs. 23.2.11,14 and 15.

Antti Karttunen, Table of n, a(n) for n = 0..16384
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 807, combined eqs. 23.2.11,14 and 15.

Cf. A060054, A006232 with A006233.

Cf. A006953, A006863, A079612, A242179, A106831, A000142, A300711.

a075180 n = a075180_list !! n
a075180_list = map (denominator . sum) $ zipWith (zipWith (%))
   (zipWith (map . (*)) a000142_list a242179_tabf) a106831_tabf
-- Reinhard Zumkeller, Jul 04 2014

a := n -> denom(bernoulli(n+1,1)/(n+1)); # Peter Luschny, Apr 22 2009

a[m_] := Sum[(-2)^(-k-1) k! StirlingS2[m,k],{k,0,m}]/(2^(m+1)-1); Table[Denominator[a[i]], {i,0,20}] (* Peter Luschny, Apr 29 2009 *)
Table[Denominator[Zeta[-n]], {n, 0, 49}] (* Alonso del Arte, Jan 13 2012 *)
CoefficientList[ Series[ EulerGamma - HarmonicNumber[n] + Log[n], {n, Infinity, 48}], 1/n] // Rest // Denominator (* Jean-François Alcover, Mar 28 2013 *)
With[{nn=50},Denominator[CoefficientList[Series[1/(1-Exp[-x])-1/x,{x,0,nn}],x] Range[0,nn-1]!]] (* Harvey P. Dale, Apr 13 2016 *)

x='x+O('x^66);
egf = 1/(1-exp(-x)) - 1/x;
v=Vec(serlaplace(egf));
vector(#v,n, denominator(v[n]))
/* Joerg Arndt, Mar 28 2013 */

A075180(n) = denominator(bernfrac(n+1)/(n+1)); \\ Antti Karttunen, Dec 19 2018, after Maple-program.

a(n) = denominator(-Zeta(-n)) = denominator(((-1)^(n+1))*B(n+1)/(n+1)), n >= 0, with Riemann's zeta function and the Bernoulli numbers B(n).
a(n) = denominators from e.g.f. (B(-x) - 1)/x, with B(x) = x/(exp(x) - 1), e.g.f. for Bernoulli numbers A027641(n)/A027642(n), n >= 0.
From Jianing Song, Apr 05 2021: (Start)
a(2n-1) = A006863(n)/2 for n > 0. By the comments in A006863, A006863(n) = A079612(2n) for n > 0. Hence a(n) = A079612(n+1)/2 all odd n. For all even n > 0, we have a(n) = 1, which is also equal to A079612(n+1)/2.
For odd n, a(n) is the product of p^(e+1) where p^e*(p-1) divides n+1 but p^(e+1)*(p-1) does not. For example, a(11) = 2^3 * 3^2 * 5^1 * 7^1 * 13^1 = 32760.
a(2n-1) = A002445(n)*(2n)/A300711(n), n > 0. (End)
a(2*n-1) = A006953(n) for n >= 1. - Georg Fischer, Dec 01 2022

More terms from Antti Karttunen, Dec 19 2018

A185633 For odd n, a(n) = 2; for even n, a(n) = denominator of Bernoulli(n)/n; The number 2 alternating with the elements of A006953.

Original entry on oeis.org

2, 12, 2, 120, 2, 252, 2, 240, 2, 132, 2, 32760, 2, 12, 2, 8160, 2, 14364, 2, 6600, 2, 276, 2, 65520, 2, 12, 2, 3480, 2, 85932, 2, 16320, 2, 12, 2, 69090840, 2, 12, 2, 541200, 2, 75852, 2, 2760, 2, 564, 2, 2227680, 2, 132, 2, 6360

Offset: 1

Paul Curtz, Dec 18 2012

nonn

There is an integer sequence b(n) = A053657(n)/2^(n-1) = 1, 1, 6, 6, 360, 360, 45360, 45360, 5443200, 5443200,... which consists of the duplicated entries of A202367.

The ratios of this sequence are b(n+1)/b(n) = 1, 6, 1, 60, 1, 126 .... = a(n)/2, which is a variant of A036283.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A006953, A007395 (bisections).

Cf. A006863, A027760, A067513, A322312, A322315 (rgs-transform).

A185633 := proc(n)
    A053657(n+1)/A053657(n) ;
end proc: # R. J. Mathar, Dec 19 2012

max = 52; s = Expand[Normal[Series[(-Log[1-x]/x)^z, {x, 0, max}]]]; a[n_, k_] := Denominator[Coefficient[s, x^n*z^k]]; A053657 = Prepend[LCM @@@ Table[a[n, k], {n, max}, {k, n}], 1]; a[n_] := A053657[[n+1]]/A053657[[n]]; Table[a[n], {n, 1, max}] (* Jean-François Alcover, Dec 20 2012 *)

A185633(n) = if(n%2,2,denominator(bernfrac(n)/(n))); \\ Antti Karttunen, Dec 03 2018

A185633(n) = { my(m=1); fordiv(n, d, if(isprime(1+d), m *= (1+d)^(1+valuation(n,1+d)))); (m); }; \\ Antti Karttunen, Dec 03 2018

a(n) = A053657(n+1)/A053657(n).
a(2*n) = 2*A036283(n).
From Antti Karttunen, Dec 03 2018: (Start)
a(n) = Product_{d|n} [(1+d)^(1+A286561(n,1+d))]^A010051(1+d) - after Peter J. Cameron's Mar 25 2002 comment in A006863.
A007947(a(n)) = A027760(n)
A001221(a(n)) = A067513(n).
A181819(a(n)) = A322312(n).
(End)

Name edited by Antti Karttunen, Dec 03 2018

A115000 a(n) = J_2(n) / 24.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 4, 7, 6, 8, 8, 12, 9, 15, 12, 16, 15, 22, 16, 25, 21, 27, 24, 35, 24, 40, 32, 40, 36, 48, 36, 57, 45, 56, 48, 70, 48, 77, 60, 72, 66, 92, 64, 98, 75, 96, 84, 117, 81, 120, 96, 120, 105, 145, 96, 155, 120, 144, 128, 168, 120, 187, 144, 176, 144, 210, 144

Offset: 5

Franklin T. Adams-Watters, Dec 10 2005

nonn

The Jordan function J_m(n) can be defined as multiplicative with J_m(p^e) = (p^m-1)*p^(m*(e-1)). Cf. A059379.
Looking at the sequences J_m(n) for fixed m, one is struck by the fact that all but a few early terms have a common factor, given in A079612. I will refer to this sequence as K(n), following the notation in the paper by Vaughan and Wooley. (The alternate lambda^*(n) in the comment for A006863 is too awkward.)
In fact, K(m) not only divides J_m(n) for all but finitely many n; it also divides Sum_{k=1..n} J_m(k) for all but finitely many n.
J_1(n) = phi(n) and phi(n)/2 and Sum_{k=1..n} phi(n)/2 are A023022 and A046657.
The weight of the n-th elliptic division polynomial -- the analog of cyclotomic polynomials for elliptic divisibility sequences. That is, let e1 = b1, e2 = b2*b1, e3 = b3*b1, e4 = b2*b4*b1, e5 = (b2^4*b4 - b3^3)*b1 = b5*e1 and so on be an elliptic divisibility sequence. Let c2 = b2^4*b4, c3 = b3^3, c4 = b4^2 and cn = bn for n>4. Then c5 = c2 - c3, c6 = c5 - c4, c7 = c6*c3 - c5*c4 and so on. Let the weight of c2, c3, c4 each be 1 and weight of a product is sum of the weights of the factors. The weight of cn is a(n) for n>4. - Michael Somos, Aug 12 2008

			G.f.: x^5 + x^6 + 2*x^7 + 2*x^8 + 3*x^9 + 3*x^10 + 5*x^11 + 4*x^12 + 7*x^13 + ...

G. C. Greubel, Table of n, a(n) for n = 5..5000
Index to divisibility sequences

Cf. A007434.

function a(n) return n lt 5 select 0 else Dimension( ModularForms( Gamma1(n), 2)) - Dimension( ModularForms( Gamma1(n), 1)); end function; /* Michael Somos, Aug 05 2014 */

a[n_] := DivisorSum[n, #^2*MoebiusMu[n/#]&]/24; Table[a[n], {n, 5, 80}] (* Jean-François Alcover, Dec 07 2015, adapted from PARI *)

{a(n) = if( n<5, 0, sumdiv(n, d, d^2 * moebius(n / d)) / 24)}; /* Michael Somos, Aug 12 2008 */

A007434(n) = 24 * a(n) unless n<5. - Michael Somos, Aug 12 2008

More terms from Michael Somos, Aug 12 2008

A281959 a(n) = sigma_25(n), the sum of the 25th powers of the divisors of n.

Original entry on oeis.org

1, 33554433, 847288609444, 1125899940397057, 298023223876953126, 28430288877251865252, 1341068619663964900808, 37778932988857102106625, 717897987692699877379693, 10000000298023223910507558, 108347059433883722041830252, 953962194872104906760006308

Offset: 1

Seiichi Manyama, Feb 03 2017

For k > 0, Sum_{n>=1} sigma_(4*k+1)(n) / exp(2*Pi*n) = Bernoulli(4*k+2)/(8*k+4). For k = 0, Sum_{n>=1} sigma(n)/exp(2*Pi*n) = 1/24 - 1/(8*Pi) = Bernoulli(2)/4 - 1/(8*Pi). - Vaclav Kotesovec, May 07 2023

Since n^25 == n^5 (mod 25), it follows that sigma_25(n) == sigma_5(n) (mod 25). In fact, sigma_25(n) == sigma_(5) (mod 13200), where 13200 = (2^4)*3*(5^2)*11 = A006863(10). - Peter Bala, Jan 12 2025

			For n = 6: The divisors of 6 are 1, 2, 3, 6, so a(6) = sigma_25(6) = 1^25 + 2^25 + 3^25 + 6^25 = 28430288877251865252. - _Felix Fröhlich_, Feb 03 2017

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Peter Luschny, Is this a new representation of (some) Bernoulli numbers?, Mathematics Stack Exchange.
Index entries for sequences related to sigma(n).

Cf. A000005, A037947, A275055, A362870.

DivisorSigma[25,Range[20]] (* Harvey P. Dale, Jul 08 2024 *)

a(n) = sigma(n, 25) \\ Felix Fröhlich, Feb 03 2017

from sympy import divisor_sigma
def A281959(n): return divisor_sigma(n,25) # Chai Wah Wu, May 07 2023

G.f.: Sum_{k>=1} k^25*x^k/(1-x^k).
a(n) == A037947(n) mod 657931.
a(n) = Sum_{k=1..A000005(n)} A275055(k)^25. - Felix Fröhlich, Feb 03 2017
Sum_{n>=1} a(n)/exp(2*Pi*n) = 657931/24 = Bernoulli(26)/52. - Vaclav Kotesovec, May 07 2023
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(25*e+25)-1)/(p^25-1).
Dirichlet g.f.: zeta(s)*zeta(s-25).
Sum_{k=1..n} a(k) = zeta(26) * n^26 / 26 + O(n^27). (End)

A345262 a(n) is the order of the image of the J-homomorphism in the stable homotopy groups of spheres.

Original entry on oeis.org

1, 2, 1, 24, 1, 1, 1, 240, 2, 2, 1, 504, 1, 1, 1, 480, 2, 2, 1, 264, 1, 1, 1, 65520, 2, 2, 1, 24, 1, 1, 1, 16320, 2, 2, 1, 28728, 1, 1, 1, 13200, 2, 2, 1, 552, 1, 1, 1, 131040, 2, 2, 1, 24, 1, 1, 1, 6960, 2, 2, 1, 171864, 1, 1, 1, 32640, 2, 2, 1, 24, 1, 1, 1

Offset: 0

Tom Harris, Jun 12 2021

Im(J) is a finite cyclic subgroup of Pi_n^S and has known order a(n) calculated by Adams using the Adams conjecture, subsequently proven by Quillen. When n is 3 or 7 mod 8 the value a(n) is related to the Bernoulli numbers; the other values of a(n) are 8-periodic (after an exceptional n=0).

D. Ravenel, Complex cobordism and stable homotopy groups of spheres (2ed), AMS Chelsea Publishing, (2003), ISBN: 978-0-8218-2967-7.

J.F. Adams, On the groups J(x), IV, Topology 5 (1966), 21-71.
D.G. Quillen, The Adams conjecture, Topology 10 (1971), 1-10.

Cf. A006863, A079612. Divides A048648.

from sympy import bernoulli
def a(n):
    if n == 0:
        return 1
    n_ = n % 8
    d = {0:2, 1:2, 2:1, 4:1, 5:1, 6:1}
    if n_ in [3, 7]:
        k = (n+1)//4
        return (bernoulli(2*k)/(4*k)).denominator
    else:
        return d[n_]

a(n) is:
                 2 if n = 0 or 1 mod 8 (except a(0) = 1)
                 1 if n = 2, 4, 5 or 6 mod 8
  A006863((n+1)/4) if n = 3 or 7 mod 8.
(A006863(k) = denominator of B_2k/4k, where B_m are the Bernoulli numbers.)

A345267 Conjectural order of the torsion subgroup of the group K_n(Z) (the algebraic K-theory groups of the integers).

Original entry on oeis.org

1, 2, 2, 48, 1, 1, 1, 240, 1, 2, 2, 1008, 1, 1, 1, 480, 1, 2, 2, 528, 1, 1, 691, 65520, 1, 2, 2, 48, 1, 1, 3617, 16320, 1, 2, 87734, 57456, 1, 1, 174611, 13200, 1, 2, 155366, 1104, 1, 1, 236364091, 131040, 1, 2, 1315862, 48, 1, 1, 3392780147, 6960, 1, 2

Offset: 0

Tom Harris, Jun 12 2021

a(n) is known for n != 0 mod 4 and is related to the Bernoulli numbers via the Riemann zeta function. See Section VI.9 of Weibel's K-book.
K_0(Z) = Z is classical, so a(0) = 1. Rognes proved that K_4(Z) = 0 in 2000 so a(4) = 1.
Otherwise the value of a(4i) = 1 is conjectural. Kurihara observed that this follows from the Kummer-Vandiver conjecture (and in fact is equivalent with it). The Kummer-Vandiver conjecture has been verified for primes up to 163 million, from which it follows that a(4i) must be at least that large if it is not 1.

C. Weibel, The K-book: An Introduction to Algebraic K-theory. Graduate Studies in Mathematics, 145. American Mathematical Society, Providence, RI, 2013. ISBN: 978-0-8218-91322.

M. Kurihara, Some remarks on conjectures about cyclotomic fields and K-groups of Z, Compositio Math. 81 (1992), 223-236.
J. Rognes, K_4(Z) is the trivial group, Topology 30 (2000), 267-281.
C. Weibel, The K-book: An Introduction to Algebraic K-theory. Graduate Studies in Mathematics, 145. American Mathematical Society, Providence, RI, 2013. ISBN: 978-0-8218-91322.

Cf. A345225 (the order of the 2-primary subgroup, divides a(n)).

A000367 / A006863 (numerator / denominator of B_2n/4n, where B_m are the Bernoulli numbers).

from sympy import bernoulli
def a(n):
    n_ = n % 8
    k = n//4 + 1
    b = bernoulli(2*k)/(4*k)
    d = {0:1, 1:2, 4:1, 5:1}
    if n_ == 2:
        return 2 * b.numerator
    elif n_ == 3:
        return 2 * b.denominator
    elif n_ == 6:
        return -1 * b.numerator
    elif n_ == 7:
        return b.denominator
    else:
        return d[n_]

Let k be the integer part of 1 + n/4.
a(n) is:
              1 if n = 0 mod 8
              2 if n = 1 mod 8
   2*A000367(k) if n = 2 mod 8
   2*A006863(k) if n = 3 mod 8
              1 if n = 4 mod 8
              1 if n = 5 mod 8
  -1*A000367(k) if n = 6 mod 8
     A006863(k) if n = 7 mod 8.

cp's OEIS Frontend

A075180 Denominators from e.g.f. 1/(1-exp(-x)) - 1/x.

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A185633 For odd n, a(n) = 2; for even n, a(n) = denominator of Bernoulli(n)/n; The number 2 alternating with the elements of A006953.

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A115000 a(n) = J_2(n) / 24.

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A281959 a(n) = sigma_25(n), the sum of the 25th powers of the divisors of n.

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A345262 a(n) is the order of the image of the J-homomorphism in the stable homotopy groups of spheres.

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A345267 Conjectural order of the torsion subgroup of the group K_n(Z) (the algebraic K-theory groups of the integers).

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