A006954 -id:A006954 - cp's OEIS frontend

A060479 Number of orbits of length n in map whose periodic points come from A006954.

6, 12, 12, 0, 12, 444, 0, 60, 84, 24, 12, 0, 0, 60, 948, 0, 0, 106428, 0, 660, 84, 24, 12, 1800, 0, 60, 0, 0, 12, 1892316, 0, 0, 1956, 0, 132, 3838380, 0, 0, 84, 5400, 12, 80928, 0, 1380, 5712, 24, 0, 92820, 0, 660, 84, 0, 12, 3838380, 24, 29820, 0, 24, 0

Offset: 1

Thomas Ward

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

A006954.

If b(n) is the (n+2)-nd term of A006954, then a(n) = (1/n)* Sum_{d|n} mu(d) * b(n/d).

More terms from Sean A. Irvine, Nov 23 2022

A238087 A129779(n+2)/A006954(n).

Original entry on oeis.org

-1, 1, -1, 1, -5, 63, -315, 99, -675675, 135135, -1640925, 83329155, -4583103525, 5791834125, -71152682225625, 14230536445125, -26797763435625, 24833960277501375, -73881031825566590625, 8546099690638125, -106610328924292590271875

Offset: 0

Paul Curtz, Feb 24 2014

sign

In A129779 or A097801 there is no mention of a link with the Bernoulli numbers.

It appears that a(n+6) is divisible by 4*(n+4)^2-1 and that the sum of the digits of a(n+5) is a multiple of 9.

			a(0) = -1/1, a(1) = 2/2, a(2) = -6/6, a(3) = 30/30, a(4) = -210/42 =-5.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000466.

a[0] = -1; a[1] = 1; a[n_] := (-1)^(n+1)*(2*n)!/(2^(n-1)*n!*Denominator[BernoulliB[2*n-2]]); Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 24 2014 *)

a(n+2) = -A004193(n+1)/A000367(n+1).

More terms from Jean-François Alcover, Feb 24 2014

A249134 Numbers k such that Bernoulli number B_k has denominator 2730.

Original entry on oeis.org

12, 24, 1308, 1884, 2004, 2364, 2532, 2724, 3804, 4008, 4044, 4188, 4236, 4668, 5052, 5064, 5268, 5388, 5484, 6252, 6492, 6564, 6756, 6852, 7044, 7188, 7356, 7404, 7608, 7764, 8124, 8412, 8472, 8796, 9084, 9228, 9852, 9876, 9924

Offset: 1

Jean-François Alcover and Paul Curtz, Oct 22 2014

nonn

2730 = 2 * 3 * 5 * 7 * 13.

			BernoulliB(12) is -691/2730, hence 12 is in the sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000367, A002445, A006954, A027642, A027760, A027762, A051222, A051225-A051230, A245056, A271634, A271635, A272138, A272139, A272140, A272183, A272184, A272185, A272186.

Reap[For[n = 0, n <= 10^4, n = n+12, If[Denominator[BernoulliB[n]] == 2730, Print[n]; Sow[n]]]][[2, 1]]
Select[Table[n, {n, 2, 10000}], Denominator@BernoulliB[#]==2730 &] (* Vincenzo Librandi, Apr 02 2015 *)

is(n)=denominator(bernfrac(n))==2730 \\ Charles R Greathouse IV, Oct 22 2014

is(n)=if(n%12 || n%16==0 || n%9==0, return(0)); forprime(p=5,107, if(n%p==0, return(0))); fordiv(n,d, if(isprime(d+1) && d>13, return(0))); 1 \\ Charles R Greathouse IV, Oct 22 2014

A002195 Numerators of coefficients for numerical integration.

Original entry on oeis.org

1, -1, 11, -191, 2497, -14797, 92427157, -36740617, 61430943169, -23133945892303, 16399688681447, -3098811853954483, 312017413700271173731, -69213549869569446541, 53903636903066465730877, -522273861988577772410712439, 644962185719868974672135609261

Offset: 0

N. J. A. Sloane

The numerators of these coefficients for numerical integration are a combination of the Bernoulli numbers B{2k}, the central factorial numbers A008955(n, k) and the factor (2n+1)!. - Johannes W. Meijer, Jan 27 2009

These numbers are the numerators of the constant term in the Laurent expansion of the cosech^(2n)(x)/2^(2n) function. - Istvan Mezo, Apr 21 2023

			a(1) = -1 because (1/3)*int(t*(t^2-1^2),t=0..1) = -1/12.
a(3) = numer((-((1/6)/2)*(4) +((-1/30)/4)*(5) - ((1/42)/6)*(1))/5!) so a(3) = -191. - _Johannes W. Meijer_, Jan 27 2009

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Iaroslav V. Blagouchine, Three notes on Ser's and Hasse's representation for the zeta-functions, Integers (2018) 18A, Article #A3.
H. E. Salzer, Coefficients for numerical integration with central differences, Phil. Mag., 35 (1944), 262-264. [Annotated scanned copy]
H. E. Salzer, XXXII. Coefficients for numerical integration with central differences, Phil. Mag., 35 (1944), 262-264.
H. E. Salzer, Coefficients for repeated integration with central differences, Journal of Mathematics and Physics, 28 (1949), 54-61.
T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 545.

Cf. A002196.

See A000367, A006954, A008955 and A009445 for underlying sequences.

a:=n->numer((2/(2*n+1)!)*int(t*product(t^2-k^2,k=1..n),t=0..1)): seq(a(n),n=0..16); # Emeric Deutsch, Feb 20 2005
nmax:=16: with(combinat): A008955 := proc(n, k): sum((-1)^j*stirling1(n+1, n+1-k+j) * stirling1(n+1, n+1-k-j), j = -k..k) end proc: Omega(0) := 1: for n from 1 to nmax do Omega(n) := sum((-1)^(k1+n+1)*(bernoulli(2*k1)/(2*k1))* A008955(n-1,n-k1), k1=1..n)/(2*n-1)! end do: a := n-> numer(Omega(n)): seq(a(n), n=0..nmax); # Johannes W. Meijer, Jan 27 2009, Revised Sep 21 2012

a[0] = 1; a[n_] := Sum[Binomial[2*n+k-1, 2*n-1]*Sum[Binomial[k, j]*Sum[(2*i-j)^(2*n+j)*Binomial[j, i]*(-1)^(-i), {i, 0, j/2}]/(2^j*(2*n+j)!), {j, 1, k}], {k, 1, 2*n}]; Table[a[n] // Numerator, {n, 0, 16}] (* Jean-François Alcover, Apr 18 2014, after Vladimir Kruchinin *)
a[n_] := Numerator[SeriesCoefficient[1/2^(2*n)*Csch[x]^(2*n), {x, 0, 0}]] (* Istvan Mezo, Apr 21 2023 *)

a(n):=num(sum(binomial(2*n+k-1,2*n-1)*sum((binomial(k,j)*sum((2*i-j)^(2*n+j)*binomial(j,i)*(-1)^(-i),i,0,j/2))/(2^j*(2*n+j)!),j,1,k),k,1,2*n)); /* Vladimir Kruchinin, Feb 04 2013 */

a(n) = numerator of (2/(2*n+1)!)*Integral_{t=0..1} t*Product_{k=1..n} t^2-k^2. - Emeric Deutsch, Jan 25 2005
a(0) = 1; a(n) = numerator [sum((-1)^(k+n+1) * (B{2k}/(2*k)) * A008955(n-1, n-k), k = 1..n)/(2*n-1)!] for n >= 1. - Johannes W. Meijer, Jan 27 2009
a(n) = numerator(sum(k=1..2*n, binomial(2*n+k-1,2*n-1)*sum(j=1..k, (binomial(k,j)*sum(i=0,j/2, (2*i-j)^(2*n+j)*binomial(j,i)*(-1)^(-i)))/(2^j*(2*n+j)!)))), n>0, a(0)=1. - Vladimir Kruchinin, Feb 04 2013

More terms from Emeric Deutsch, Jan 25 2005

Edited by Johannes W. Meijer, Sep 21 2012

A002196 Denominators of coefficients for numerical integration.

Original entry on oeis.org

1, 12, 720, 60480, 3628800, 95800320, 2615348736000, 4483454976000, 32011868528640000, 51090942171709440000, 152579284313702400000, 120866571766215475200000, 50814724101952310083584000000

Offset: 0

N. J. A. Sloane

The denominators of these coefficients for numerical integration are a combination of the Bernoulli numbers B{2k}, the central factorial numbers A008955(n, k) and the factor (2n+1)!. - Johannes W. Meijer, Jan 27 2009

These numbers are the denominators of the constant term in the Laurent expansion of the even powers of the hyperbolic cosecant cosech^(2n)(x)/2^(2n) function. - Istvan Mezo, Apr 21 2023

			a(1) = 12 because (1/3)*int(t*(t^2-1^2), t=0..1) = -1/12.
a(3) = denom((-((1/6)/2)*(4) +((-1/30)/4)*(5) - ((1/42)/6)*(1))/5!) so a(3) = 60480. - _Johannes W. Meijer_, Jan 27 2009

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Iaroslav V. Blagouchine, Three notes on Ser's and Hasse's representation for the zeta-functions, Integers (2018) 18A, Article #A3.
H. E. Salzer, Coefficients for numerical integration with central differences, Phil. Mag., 35 (1944), 262-264. [Annotated scanned copy]
H. E. Salzer, XXXII. Coefficients for numerical integration with central differences, Phil. Mag., 35 (1944), 262-264.
H. E. Salzer, Coefficients for repeated integration with central differences, Journal of Mathematics and Physics, 28 (1949), 54-61.
T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 545.

Cf. A002195.
See A000367, A006954, A008955 and A009445 for underlying sequences.
Factor of ZS1[ -1,n] matrix coefficients in A160474.

a := n->denom((2/(2*n+1)!)*int(t*product(t^2-k^2,k=1..n),t=0..1)): seq(a(n), n=0..14); # Emeric Deutsch, Feb 20 2005
nmax:=12: with(combinat): A008955 := proc(n, k): sum((-1)^j*stirling1(n+1, n+1-k+j) * stirling1(n+1, n+1-k-j), j = -k..k) end proc: Omega(0) := 1: for n from 1 to nmax do Omega(n) := sum((-1)^(k1+n+1)*(bernoulli(2*k1)/(2*k1)) * A008955(n-1,n-k1), k1=1..n)/(2*n-1)! end do: a := n-> denom(Omega(n)): seq(a(n), n=0..nmax); # Johannes W. Meijer, Jan 27 2009, Revised Sep 21 2012

a[0] = 1; a[n_] := Sum[Binomial[2*n+k-1, 2*n-1]*Sum[Binomial[k, j]*Sum[(2*i-j)^(2*n+j)*Binomial[j, i]*(-1)^(-i), {i, 0, j/2}]/(2^j*(2*n+j)!), {j, 1, k}], {k, 1, 2*n}]/2^(2*n-1); Table[a[n] // Denominator, {n, 0, 12}] (* Jean-François Alcover, Apr 18 2014, after Vladimir Kruchinin *)
a[n_] := Denominator[SeriesCoefficient[1/2^(2*n)*Csch[x]^(2*n), {x, 0, 0}]] (* Istvan Mezo, Apr 21 2023 *)

a(n) = denominator of (2/(2*n+1)!)*int(t*product(t^2-k^2, k=1..n), t=0..1). - Emeric Deutsch, Jan 25 2005

a(0) = 1; a(n) = denominator [sum((-1)^(k+n+1) * (B{2k}/(2*k)) * A008955(n-1, n-k), k = 1..n) / (2*n-1)!] for n >= 1. - Johannes W. Meijer, Jan 27 2009

More terms from Emeric Deutsch, Jan 25 2005

Edited by Johannes W. Meijer, Sep 21 2012

A027762 Denominator of Sum_{p prime, p-1 divides 2*n} 1/p.

Original entry on oeis.org

6, 30, 42, 30, 66, 2730, 6, 510, 798, 330, 138, 2730, 6, 870, 14322, 510, 6, 1919190, 6, 13530, 1806, 690, 282, 46410, 66, 1590, 798, 870, 354, 56786730, 6, 510, 64722, 30, 4686, 140100870, 6, 30, 3318, 230010, 498, 3404310, 6, 61410, 272118, 1410, 6, 4501770

Offset: 1

N. J. A. Sloane

From the von Staudt-Clausen theorem, denominator(B_2n) = product of primes p such that (p-1)|2n.

Same as A002445.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Th. 118.
H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.

R. Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.
Index entries for sequences related to Bernoulli numbers.

Cf. A027761, A006954.

Cf. A027760.

a(n)=
{
    my(s=0);
    forprime (p=2, 2*n+1, if( (2*n)%(p-1)==0, s+=1/p ) );
    return( denominator(s) );
}
/* Joerg Arndt, May 06 2012 */

a(n) = A002445(n). [Joerg Arndt, May 06 2012]

a(n) = A027760(2*n). - Ridouane Oudra, Feb 22 2022

A002197 Numerators of coefficients for numerical integration.

Original entry on oeis.org

1, 17, 367, 27859, 1295803, 5329242827, 25198857127, 11959712166949, 11153239773419941, 31326450596954510807, 3737565567167418110609, 2102602044094540855003573, 189861334343507894443216783

Offset: 0

N. J. A. Sloane

nonn

The numerators of these coefficients for numerical integration are a combination of the Bernoulli numbers B{2k}, the central factorial numbers A008956(n, k) and the factor 4^n*(2*n+1)!. - Johannes W. Meijer, Jan 27 2009

			a(2) = numer(((1-2^1)*(-1)*((1/6)/2)*(9) + (1-2^3)*(1)*((-1/30)/4)*(10) + (1-2^5)*(-1)*((1/42)/6)*(1))/(2*4^2*5!)) so a(2) = 367. - _Johannes W. Meijer_, Jan 27 2009

H. E. Salzer, Coefficients for mid-interval numerical integration with central differences, Phil. Mag., 36 (1945), 216-218.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..100
H. E. Salzer, Coefficients for mid-interval numerical integration with central differences, Phil. Mag., 36 (1945), 216-218. [Annotated scanned copy]
T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 545.

Cf. A002198.
See A000367, A006954, A008956 and A002671 for underlying sequences.
Factor of the LS1[-2,n] matrix coefficients in A160487.

nmax:=13: for n from 0 to nmax do A008956(n, 0) := 1: A008956(n, n) := (doublefactorial(2*n-1))^2 od: for n from 1 to nmax do for k from 1 to n-1 do A008956(n, k) := (2*n-1)^2*A008956(n-1, k-1) + A008956(n-1, k) od: od: for n from 0 to nmax do Delta(n) := sum((1-2^(2*k1-1)) * (-1)^(k1) * (bernoulli(2*k1)/(2*k1)) * A008956(n, n+1-k1), k1=1..n+1) / (2*4^(n)*(2*n+1)!) end do: a:=n-> numer(Delta(n)): seq(a(n), n=0..nmax-1); # Johannes W. Meijer, Jan 27 2009, revised Sep 21 2012

CoefficientList[Series[1/x - 1/Sqrt[x]/ArcSin[Sqrt[x]], {x, 0, 12}], x] // Numerator (* Jean-François Alcover, Jul 05 2011, after Vladeta Jovovic *)

a(n):=(sum(binomial(2*n+k-1,2*n-2)*sum((binomial(k+1,j)*sum((2*i-j)^(2*n+j)*binomial(j,i)*(-1)^(n-i),i,0,j/2))/(2^(j-1)*(2*n+j)!),j,1,k+1),k,0,2*n-1))/(2*n-1);
makelist(num(a(n)),n,0,10); /* Vladimir Kruchinin, May 16 2013 */

Numerators of coefficients in expansion of 1/x-1/sqrt(x)/arcsin(sqrt(x)). - Vladeta Jovovic, Aug 11 2002
a(n) = numerator [sum((1-2^(2*k-1)) * (-1)^(k) * (B{2k}/(2*k)) * A008956(n, n+1-k), k=1..n+1) / (2*4^(n)*(2*n+1)!)] for n >= 0. - Johannes W. Meijer, Jan 27 2009
a(n) = numerator((sum(k=0..2*n-1, binomial(2*n+k-1,2*n-2)*sum(j=1..k+1, (binomial(k+1,j)*sum(i=0..j/2,(2*i-j)^(2*n+j)*binomial(j,i)*(-1)^(n-i)))/(2^(j-1)*(2*n+j)!))))/(2*n-1)). - Vladimir Kruchinin, May 16 2013

More terms from Vladeta Jovovic, Aug 11 2002

Edited by Johannes W. Meijer, Sep 21 2012

A002198 Denominators of coefficients for numerical integration.

Original entry on oeis.org

24, 5760, 967680, 464486400, 122624409600, 2678117105664000, 64274810535936000, 149852129706639360000, 669659197233029971968000, 8839501403475995629977600000, 4879404774718749587747635200000

Offset: 0

N. J. A. Sloane

nonn

The denominators of these coefficients for numerical integration are a combination of the Bernoulli numbers B{2k}, the central factorial numbers A008956(n, k) and the factor 4^n*(2*n+1)!. - Johannes W. Meijer, Jan 27 2009

H. E. Salzer, Coefficients for mid-interval numerical integration with central differences, Phil. Mag., 36 (1945), 216-218.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
H. E. Salzer, Coefficients for mid-interval numerical integration with central differences, Phil. Mag., 36 (1945), 216-218. [Annotated scanned copy]
T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 545.

Cf. A002197.
See A000367, A006954, A008956 and A002671 for underlying sequences.
Factor of the LS1[ -2,n] matrix coefficients in A160487.

nmax:=10: for n from 0 to nmax do A008956(n, 0) := 1: A008956(n, n) := (doublefactorial(2*n-1))^2 od: for n from 1 to nmax do for k from 1 to n-1 do A008956(n, k) := (2*n-1)^2*A008956(n-1, k-1) + A008956(n-1, k) od: od:
for n from 0 to nmax do Delta(n) := add((1-2^(2*k1-1)) * (-1)^k1 * (bernoulli(2*k1)/(2*k1)) * A008956(n, n+1-k1), k1=1..n+1) / (2*4^(n)*(2*n+1)!) end do: a:=n-> denom (Delta(n)): seq(a(n), n=0..nmax); # Johannes W. Meijer, Jan 27 2009, Revised Sep 21 2012

a(n) = denominator(Sum_{k=1..n+1}((1-2^(2*k-1))*(-1)^k*(B_{2k}/(2*k))*A008956(n, n+1-k)) / (2*4^(n)*(2*n+1)!)) for n >= 0. - Johannes W. Meijer, Jan 27 2009

Two more terms and editing by Johannes W. Meijer, Sep 21 2012

A166120 a(n) = A027642(n-1) / A089026(n).

Original entry on oeis.org

1, 1, 2, 1, 6, 1, 6, 1, 30, 1, 6, 1, 210, 1, 6, 1, 30, 1, 42, 1, 330, 1, 6, 1, 2730, 1, 6, 1, 30, 1, 462, 1, 510, 1, 6, 1, 51870, 1, 6, 1, 330, 1, 42, 1, 690, 1, 6, 1, 46410, 1, 66, 1, 30, 1, 798, 1, 870, 1, 6, 1, 930930, 1, 6, 1, 510, 1, 966, 1, 30, 1, 66, 1, 1919190, 1, 6, 1, 30, 1, 42, 1

Offset: 1

Paul Curtz, Oct 07 2009

nonn

As in A166062, the offset is rather arbitrary.
The sequence contains numbers like 210 which are not in A006954.
One could also consider dividing by the largest prime divisor of A027642 instead of A089026, which yields 1, 1, 2, 1, 6, 1, 6, 1, 6, 1, 6, 1, 210, 1, 2, 1, 30, 1, 42, 1, 30, ... as an alternative version.
These are the Clausen numbers based on the proper divisors of n whereas the classical Clausen numbers A160014 are based on all divisors of n. (The proper divisors are the divisors of n that are less than n.) - Peter Luschny, Aug 20 2022

Cf. A027642, A089026, A160014.

A027642 := proc(n) denom(bernoulli(n)) ; end:
A089026 := proc(n) if isprime(n) then n; else 1; end if; end proc:
A166120 := proc(n) A027642(n-1)/A089026(n) ; end proc: seq(A166120(n), n=1..80) ; # R. J. Mathar, Mar 25 2010
# Second program, assuming offset 0:
clausen := proc(n) if irem(n,2)=1 then 1 else numtheory[divisors](n) minus {n};
map(i -> i+1, %); select(isprime, %); mul(i, i=%) fi end:
seq(clausen(n), n = 0..79); # Peter Luschny, Aug 20 2022

Extended by R. J. Mathar, Mar 25 2010

A174276 Denominator of the n-th term of the inverse binomial transform of 1, 1/2, B_4, B_6, B_8,..., a modified Bernoulli sequence.

Original entry on oeis.org

1, 2, 6, 30, 70, 210, 2310, 10010, 6006, 510510, 461890, 9699690, 31870410, 74364290, 223092870, 6469693230, 6077590610, 200560490130, 200560490130, 494715875654, 674612557710, 60850052705442, 872184088778002, 13082761331670030

Offset: 0

Paul Curtz, Mar 14 2010

The numerators are in A174289. The input sequence starts 1, 1/2, 1/6, -1/30, 1/42....

The inverse binomial transform generates 1, -1/2, 1/6, -1/30, 11/70, -137/210, 4157/2310,...

Cf. A000367, A164555, A006954.

read("transforms") ; L := [1,1/2,seq(bernoulli(2*i),i=1..30)] ; BINOMIALi(L) ; apply(denom,%) ;

b[0]=1; b[1]=1/2; b[n_] := BernoulliB[2n-2]; a[n_] := Sum[(-1)^(n-k)*Binomial[n, k]*b[k], {k, 0, n}]; Table[a[n], {n, 0, 23}] // Denominator (* Jean-François Alcover_, Aug 09 2012 *)

cp's OEIS Frontend

A060479 Number of orbits of length n in map whose periodic points come from A006954.

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A238087 A129779(n+2)/A006954(n).

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A249134 Numbers k such that Bernoulli number B_k has denominator 2730.

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A002195 Numerators of coefficients for numerical integration.

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Maple

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Maxima

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A002196 Denominators of coefficients for numerical integration.

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Maple

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A027762 Denominator of Sum_{p prime, p-1 divides 2*n} 1/p.

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PARI

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A002197 Numerators of coefficients for numerical integration.

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