A141417 -id:A141417 - cp's OEIS frontend

A027760 Denominator of Sum_{p prime, p-1 divides n} 1/p.

2, 6, 2, 30, 2, 42, 2, 30, 2, 66, 2, 2730, 2, 6, 2, 510, 2, 798, 2, 330, 2, 138, 2, 2730, 2, 6, 2, 870, 2, 14322, 2, 510, 2, 6, 2, 1919190, 2, 6, 2, 13530, 2, 1806, 2, 690, 2, 282, 2, 46410, 2, 66, 2, 1590, 2, 798, 2, 870, 2, 354, 2, 56786730, 2, 6, 2, 510, 2

Offset: 1

N. J. A. Sloane

The GCD of integers x^(n+1)-x, for all integers x. - Roger Cuculiere (cuculier(AT)imaginet.fr), Jan 19 2002
If each x in a ring satisfies x^(n+1)=x, the characteristic of the ring is a divisor of a(n) (Rosenblum 1977). - Daniel M. Rosenblum (DMRosenblum(AT)world.oberlin.edu), Sep 24 2008
The denominators of the Bernoulli numbers for n>0. B_n sequence begins 1, -1/2, 1/6, 0/2, -1/30, 0/2, 1/42, 0/2, ... This is an alternative version of A027642 suggested by the theorem of Clausen. To add a(0) = 1 has been proposed in A141056. - Peter Luschny, Apr 29 2009
For N > 1, a(n) is the greatest number k such that x*y^n ==y*x^n (mod k) for any integers x and y.  Example: a(19) = 798 because x*y^19 ==y*x^19 (mod 798). - Michel Lagneau, Apr 21 2012
a(n) is the largest k such that b^(n+1) == b (mod k) for every integer b. - Mateusz Szymański, Feb 18 2016, corrected by Thomas Ordowski, Jul 01 2018
When n is even, a(n) is the product of the distinct primes dividing the denominator of zeta(1-n), where zeta(s) is the Riemann zeta function. - Griffin N. Macris, Jun 13 2016
If n+1 is prime, then A002322(a(n)) = n. Composite numbers n+1 such that A002322(a(n)) = n are in A317210. - Max Alekseyev and Thomas Ordowski, Jul 09 2018

			1/2, 5/6, 1/2, 31/30, 1/2, 41/42, 1/2, 31/30, 1/2, 61/66, 1/2, 3421/2730, 1/2, 5/6, 1/2, 557/510, ...

T. D. Noe, Table of n, a(n) for n = 1..10000
Thomas Clausen, Lehrsatz aus einer Abhandlung Über die Bernoullischen Zahlen, Astr. Nachr. 17 (1840), 351-352. [_Peter Luschny_, Apr 29 2009]
S. C. Locke and A. Mandel, Problem E 2901, American Mathematical Monthly 88 (1981), p. 538. Solution in Vol. 90 (1983), pp. 212-213. [Daniel M. Rosenblum (DMRosenblum(AT)world.oberlin.edu), Jul 31 2008]
D. M. Rosenblum, Problem 1019, Mathematics Magazine 50 (1977), p. 164. Solution by T. Orloff in Vol. 52 (1979), p. 50.
Wikipedia, Bernoulli number

Cf. A002997, A027759, A027642, A141056, A317210.

A027760 := proc(n) local s,p; s := 0 ; p := 2; while p <= n+1 do if n mod (p-1) = 0 then s := s+1/p; fi; p := nextprime(p) ; od: denom(s) ; end: # R. J. Mathar, Aug 12 2008

clausen[n_] := Product[i, {i, Select[ Map[ # + 1 &, Divisors[n]], PrimeQ]}]
Table[clausen[i], {i, 1, 20}] (* Peter Luschny, Apr 29 2009 *)
f[n_] := Times @@ Select[Divisors@n + 1, PrimeQ]; Array[f, 56] (* Robert G. Wilson v, Apr 25 2012 *)

a(n)=denominator(sumdiv(n,d,if(isprime(d+1),1/(d+1)))) \\ Charles R Greathouse IV, Jul 08 2011

a(n)=my(pr=1);fordiv(n,d,if(isprime(d+1),pr*=d+1));pr \\ Charles R Greathouse IV, Jul 08 2011

def A027760(n):
    return mul(filter(lambda s: is_prime(s), map(lambda i: i+1, divisors(n))))
[A027760(n) for n in (1..56)]  # Peter Luschny, May 23 2013

a(2*k) = A091137(2*k)/A091137(2*k-1). - Paul Curtz, Aug 05 2008
a(n) = product_{p prime, p-1 divides n}. - Eric M. Schmidt, Aug 01 2013
a(2n-1) = 2. - Robert G. Wilson v, Jul 23 2018

Formula submitted with A141417 added by R. J. Mathar, Nov 17 2010

A140811 a(n) = 6*n^2 - 1.

Original entry on oeis.org

-1, 5, 23, 53, 95, 149, 215, 293, 383, 485, 599, 725, 863, 1013, 1175, 1349, 1535, 1733, 1943, 2165, 2399, 2645, 2903, 3173, 3455, 3749, 4055, 4373, 4703, 5045, 5399, 5765, 6143, 6533, 6935, 7349, 7775, 8213, 8663, 9125, 9599, 10085, 10583, 11093, 11615

Offset: 0

Paul Curtz, Jul 16 2008

Also: The numerators in the j=2 column of the array a(i,j) defined in A140825, where the columns j=0 and j=1 are represented by A000012 and A005408. This could be extended to column j=3: 1, -1, 9, 55, 161, ... The common feature of these sequences derived from a(i,j) is that their j-th differences are constant sequences defined by A091137(j).
a(n) is the set of all k such that 6*k + 6 is a perfect square. - Gary Detlefs, Mar 04 2010
The identity (6*n^2 - 1)^2 - (9*n^2 - 3)*(2*n)^2 = 1 can be written as a(n+1)^2 - A157872(n)*A005843(n+1)^2 = 1. - Vincenzo Librandi, Feb 05 2012
Apart from first term, sequence found by reading the line from 5, in the direction 5, 23, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Jul 18 2012
From Paul Curtz, Sep 17 2018: (Start)
Terms from center to right in the following spiral:
.
                    65--63--61--59
                    /             \
                  67  31--29--27  57
                  /   /         \   \
                69  33   9---7  25  55
                /   /   /     \   \   \
              71  35  11  -1===5==23==53==>
                  /   /   /   /   /   /
                37  13   1---3  21  51
                  \   \         /   /
                  39  15--17--19  49
                    \             /
                    41--43--45--47 (End)

P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Note 12, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969, 132 pages, pp. 28-36. CCSA, then CELAR. Now DGA Maitrise de l'Information 35131 Bruz.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Leo Tavares, Illustration: Barred Stars.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A157872, A060747, A103115(n+1), A141417 (array).
Cf. A000012, A001318, A005408, A091137, A140825.
Cf. A003154, A032528, A033581, A016777, A017593.

[6*n^2 - 1: n in [0..50]]; // Vincenzo Librandi, Jun 02 2011

LinearRecurrence[{3,-3,1},{-1,5,23},40] (* Vincenzo Librandi, Feb 05 2012 *)
CoefficientList[Series[(1-8*x-5*x^2)/(x-1)^3 , {x, 0, 40}], x] (* Stefano Spezia, Sep 17 2018 *)

a(n)=6*n^2-1 \\ Charles R Greathouse IV, Jun 01 2011

a(n) = 2*a(n-1) - a(n-2) + 12.
First differences: a(n+1) - a(n) = A017593(n).
Second differences: A071593(n+1) - A071593(n) = 12.
G.f.: (1-8*x-5*x^2)/(x-1)^3. - Jaume Oliver Lafont, Aug 30 2009
From Vincenzo Librandi, Feb 05 2012: (Start)
a(n) = a(n-1) + 12*n - 6.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
a(n) = A033581(n) - 1. - Omar E. Pol, Jul 18 2012
a(n) = A032528(2*n) - 1. - Adriano Caroli, Jul 21 2013
For n > 0, a(n) = floor(3/(cosh(1/n) - 1)) = floor(1/(n*sinh(1/n) - 1)); for similar formulas for cosine and sine, see A033581. - Clark Kimberling, Oct 19 2014, corrected by M. F. Hasler, Oct 21 2014
a(-n) = a(n). - Paul Curtz, Sep 17 2018
From Amiram Eldar, Feb 04 2021: (Start)
Sum_{n>=1} 1/a(n) = (1 - (Pi/sqrt(6))*cot(Pi/sqrt(6)))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(6))*csc(Pi/sqrt(6)) - 1)/2.
Product_{n>=1} (1 + 1/a(n)) = (Pi/sqrt(6))*csc(Pi/sqrt(6)).
Product_{n>=1} (1 - 1/a(n)) = csc(Pi/sqrt(6))*sin(Pi/sqrt(3))/sqrt(2). (End)
a(n) = A003154(n+1) - 2*A016777(n). - Leo Tavares, May 13 2022
E.g.f.: exp(x)*(6*x^2 + 6*x - 1). - Elmo R. Oliveira, Jan 16 2025

Edited and extended by R. J. Mathar, Aug 06 2008

Better description Ray Chandler, Feb 03 2009

A195189 Denominators of a sequence leading to gamma = A001620.

Original entry on oeis.org

2, 24, 72, 2880, 800, 362880, 169344, 29030400, 9331200, 4790016000, 8673280, 31384184832000, 6181733376000, 439378587648000, 10346434560000, 512189896458240000, 265423814656, 14148260909088768000, 2076423318208512000, 96342919523794944000000, 74538995631567667200000

Offset: 0

Paul Curtz, Sep 11 2011

gamma = 1/2 + 1/24 + 1/72 + 19/2880 + 3/800 + 863/362880 + 275/169344 + ... = (A002206 unsigned=reduced A141417(n+1)/A091137(n+1))/a(n) is an old formula based on Gregory's A002206/A002207.

This formula for Euler's constant was discovered circa 1780-1790 by the Italian mathematicians Gregorio Fontana (1735-1803) and Lorenzo Mascheroni (1750-1800), and was subsequently rediscovered several times (in particular, by Ernst Schröder in 1879, Niels E. Nørlund in 1923, Jan C. Kluyver in 1924, Charles Jordan in 1929, Kenter in 1999, and Victor Kowalenko in 2008). For more details, see references below. - Iaroslav V. Blagouchine, May 03 2015

			a(0)=1*2, a(1)=2*12, a(2)=3*24, a(3)=4*720.

G. C. Greubel, Table of n, a(n) for n = 0..440
Iaroslav V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations, Journal of Number Theory (Elsevier), vol. 148, pp. 537-592 and vol. 151, pp. 276-277, 2015. arXiv version.
Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in 1/pi^2 and into the formal enveloping series with rational coefficients only, Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. arXiv version.
M. Coffey and J. Sondow, Rebuttal of Kowalenko's paper as concerns the irrationality of Euler's constant, Acta Appl. Math., 121 (2012), 1-3.
J. C. Kluyver, Euler's constant and natural numbers, Proc. Kon. Ned. Akad. Wet., 27(1-2) (1924), 142-144.

Cf. A001620, A002206, A002207, A091137, A141417.

g[n_]:=Sum[StirlingS1[n,l]/(l+1),{l,1,n}]/(n*n!); a[n_]:=Denominator[g[n]]; Table[a[n],{n,1,30}] (* Iaroslav V. Blagouchine, May 03 2015 *)
g[n_] := Sum[ BernoulliB[j]/j * StirlingS1[n, j-1], {j, 1, n+1}] / n! ; a[n_] := (n+1)*Denominator[g[n]]; Table[a[n], {n, 0, 20}]
(* or *) max = 20; Denominator[ CoefficientList[ Series[ 1/Log[1 + x] - 1/x, {x, 0, max}], x]]*Range[max+1] (* Jean-François Alcover, Sep 04 2013 *)

a(n) = (n+1) * A002207(n).

More terms from Jean-François Alcover, Sep 04 2013

A141530 a(n) = 4n^3 - 6n^2 + 1.

Original entry on oeis.org

1, -1, 9, 55, 161, 351, 649, 1079, 1665, 2431, 3401, 4599, 6049, 7775, 9801, 12151, 14849, 17919, 21385, 25271, 29601, 34399, 39689, 45495, 51841, 58751, 66249, 74359, 83105, 92511, 102601, 113399, 124929, 137215, 150281, 164151, 178849, 194399, 210825, 228151

Offset: 0

Paul Curtz, Aug 12 2008

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A046092, A141047, A141417.

See Librandi's comment in A078371.

[4*n^3 -6*n^2 +1: n in [0..50]]; // G. C. Greubel, Mar 29 2021

A141530:= n-> 4*n^3 -6*n^2 +1; seq(A141530(n), n=0..50); # G. C. Greubel, Mar 29 2021

Array[4*#^3-6*#^2+1&,50,0] (* Vladimir Joseph Stephan Orlovsky, Nov 03 2009 *)
LinearRecurrence[{4,-6,4,-1},{1,-1,9,55},50] (* Harvey P. Dale, Nov 30 2011 *)

a(n)=4*n^3-6*n^2+1 \\ Charles R Greathouse IV, Oct 07 2015

def A141530(n): return (m:=(n<<1)-1)*(n*(m-1)-1) # Chai Wah Wu, Mar 11 2024

[4*n^3 -6*n^2 +1 for n in (0..50)] # G. C. Greubel, Mar 29 2021

a(n) = (2*n-1)*(2*n^2 - 2*n - 1) = A060747(n)*A132209(n-1), n > 1. - R. J. Mathar, Feb 22 2009
G.f.: (1 - 5*x + 19*x^2 + 9*x^3)/(1-x)^4. - Jaume Oliver Lafont, Aug 30 2009
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) with a(0)=1, a(1)=-1, a(2)=9, a(3)=55. - Harvey P. Dale, Nov 30 2011
E.g.f.: (1 - 2*x + 6*x^2 + 4*x^3)*exp(x). - G. C. Greubel, Mar 29 2021

Corrected, completed and edited, following an observation from Vincenzo Librandi, by M. F. Hasler, Feb 12 2009

Further edited by N. J. A. Sloane, Feb 13 2009

A007525 Decimal expansion of log_2 e.

Original entry on oeis.org

1, 4, 4, 2, 6, 9, 5, 0, 4, 0, 8, 8, 8, 9, 6, 3, 4, 0, 7, 3, 5, 9, 9, 2, 4, 6, 8, 1, 0, 0, 1, 8, 9, 2, 1, 3, 7, 4, 2, 6, 6, 4, 5, 9, 5, 4, 1, 5, 2, 9, 8, 5, 9, 3, 4, 1, 3, 5, 4, 4, 9, 4, 0, 6, 9, 3, 1, 1, 0, 9, 2, 1, 9, 1, 8, 1, 1, 8, 5, 0, 7, 9, 8, 8, 5, 5, 2, 6, 6, 2, 2, 8, 9, 3, 5, 0, 6, 3, 4, 4, 4, 9, 6, 9, 9

Offset: 1

N. J. A. Sloane

Around 1670, James Gregory discovered by inversion of 1 - 1/2 + 1/3 - 1/4 + 1/5 - ... = log(2) that 1 + 1/2 - 1/12 + 1/24 - 19/720 + (27/1440 = 3/160) - 863/60480 + ... = 1/log(2). See formula with A002206 and A002207. See also A141417 signed /A091137; case i = 0 in A165313. First row in array p. 36 of the reference. - Paul Curtz, Sep 12 2011

This constant 1/log(2) is also related to the asymptotic evaluation of the maximum number of subtraction steps required to compute gcd(m, n) by the binary Euclidean algorithm, m and n being odd and chosen at random. - Jean-François Alcover, Jun 23 2014, after Steven Finch

			1.442695040888963407359924681...

Paul Curtz, Intégration numérique des systèmes différentiels .. , note n° 12, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.18 Porter-Hensley constants, p. 159.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 25, equation 25:14:3 at page 232.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Simon Plouffe, 1/log(2) the inverse of the natural logarithm of 2.
Srinivasa Ramanujan, Question 769, J. Ind. Math. Soc.
Index entries for transcendental numbers.

Cf. A000670, A002206, A002207, A052882, A091137, A141417, A165313.

RealDigits[N[1/Log[2], 105]][[1]] (* Jean-François Alcover, Oct 30 2012 *)

1/log(2) \\ Charles R Greathouse IV, Jan 04 2016

Equals lim_{n->infinity} A000670(n)/A052882(n). - Mats Granvik, Aug 10 2009
Equals Sum_{k>=-1} A002206(k)/A002207(k). - Paul Curtz, Sep 12 2011
Also equals integral_{x>=2} 1/(x*log(x)^2). - Jean-François Alcover, May 24 2013
1/log(2) = Sum_{n = -infinity..infinity} (2^n / (1 + 2^2^n)). - Nicolas Nagel, Mar 16 2018
More generally: 1/log(2) = Sum_{n = -infinity..infinity} (2^(n+x) / (1 + 2^2^(n+x))) for all real x. - Nicolas Nagel, Jul 02 2019
From Amiram Eldar, Jun 04 2023: (Start)
Equals 1 + Sum_{k>=1} 1/(2^k * (1 + 2^(1/2^k))).
Equals Product_{k>=1} ((1 + 2^(1/2^k))/2). (End)

A165313 Triangle T(n,k) = A091137(k-1) read by rows.

Original entry on oeis.org

1, 1, 2, 1, 2, 12, 1, 2, 12, 24, 1, 2, 12, 24, 720, 1, 2, 12, 24, 720, 1440, 1, 2, 12, 24, 720, 1440, 60480, 1, 2, 12, 24, 720, 1440, 60480, 120960, 1, 2, 12, 24, 720, 1440, 60480, 120960, 3628800, 1, 2, 12, 24, 720, 1440, 60480, 120960, 3628800, 7257600, 1, 2, 12

Offset: 1

Paul Curtz, Sep 14 2009

From a study of modified initialization formulas in Adams-Bashforth (1855-1883) multisteps method for numerical integration. On p.36, a(i,j) comes from (j!)*a(i,j) = Integral_{u=i,..,i+1} u*(u-1)*...*(u-j+1) du; see p.32.
Then, with i vertical, j horizontal, with unreduced fractions, partial array is:
0) 1,  1/2,  -1/12,   1/24,  -19/720,   27/1440, ... =  1/log(2)
1) 1,  3/2,   5/12,  -1/24,   11/720,  -11/1440, ... =  2/log(2)
2) 1,  5/2,  23/12,   9/24,  -19/720,   11/1440, ... =  4/log(2)
3) 1,  7/2,  53/12,  55/24,  251/720,  -27/1440, ... =  8/log(2)
4) 1,  9/2,  95/12, 161/24, 1901/720,  475/1440, ... = 16/log(2)
5) 1, 11/2, 149/12, 351/24, 6731/720, 4277/1440, ... = 32/log(2)
... [improved by Paul Curtz, Jul 13 2019]
First line: the reduced terms are A002206/A002207, logarithmic or Gregory numbers G(n). The difference between the second line and the first one is 0 together A002206/A002207. This is valid for the next lines. - Paul Curtz, Jul 13 2019
See A141417, A140825, A157982, horizontal numerators: A141047, vertical numerators: A000012, A005408, A140811, A141530, A157411. On p.56, coefficients are s(i,q) = (1/q!)* Integral_{u=-i-1,..,1} u*(u+1)*...*(u+q-1) du.
Unreduced fractions array is:
-1) 1,   1/2,  5/12,   9/24, 251/720, 475/1440, ... = A002657/A091137
0)  2,   0/2,  4/12,   8/24, 232/720, 448/1440, ... = A195287/A091137
1)  3,  -3/2,  9/12,   9/24, 243/720, 459/1440, ...
2)  4,  -8/2, 32/12,   0/24, 224/720, 448/1440, ...
3)  5, -15/2, 85/12, -55/24, 475/720, 475/1440, ...
...
(on p.56 up to 6)). See A147998. Vertical numerators: A000027, A147998, A152064, A157371, A165281.
From Paul Curtz, Jul 14 2019: (Start)
Difference table from the second line and the first one difference:
1,                -1/2,      -1/12,    -1/24,    -19/720, -27/1440, ...
-3/2,             5/12,       1/24,    11/720,    11/1440, ...
23/12,           -9/24,    -19/720,  -11/1440, ...
-55/24,        251/720,    27/1440, ...
1901/720,    -475/1440,
-4277/1440, ...
...
Compare the lines to those of the first array.
The verticals are the signed diagonals of the first array. (End)

			1;
1,2;
1,2,12;
1,2,12,24;
1,2,12,24,720;

P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Centre de Calcul Scientifique de l'Armement, Note 12, Arcueil, 1969.

Cf. A090624, A091137.
Cf. A000012, A000079, A002657, A005408, A007525, A131920, A140811, A140825, A141047, A141417, A141530, A157411, A157982, A195287.
Cf. A002206, A002207.

(* a = A091137 *) a[n_] := a[n] = Product[d, {d, Select[Divisors[n]+1, PrimeQ]}]*a[n-1]; a[0]=1; Table[Table[a[k-1], {k, 1, n}], {n, 1, 11}] // Flatten (* Jean-François Alcover, Dec 18 2014 *)

A193546 Numerator of the third row of the inverse Akiyama-Tanigawa algorithm from 1/n.

Original entry on oeis.org

1, 1, 7, 17, 41, 731, 8563, 27719, 190073, 516149, 1013143139, 1519024289, 14108351869, 14399405173, 23142912688967, 83945247395407, 84894728616107, 3204549982389941, 262488267575333123, 9027726081126601799, 2026692221793223022131, 1375035304877251309001

Offset: 0

Paul Curtz, Aug 27 2011

Akiyama-Tanigawa from 1/n gives Bernoulli A164555(n)/A027642(n).
Reciprocally
1,   1/2,   5/12,     3/8, 251/720,    95/288, 19087/60480, 5257/17280,
1/2, 1/6,    1/8,  19/180,    3/32, 863/10080,    275/3456,
1/3, 1/12, 7/120,  17/360, 41/1008, 731/20160, 8563/259200,
1/4, 1/20, 1/30,   11/420, 89/4032,5849/302400,
1/5, 1/30, 3/140, 83/5040, 59/4320,
1/6, 1/42, 5/336,
1/7, 1/56,
1/8.
First row: A002208/A002209 or reduced A002657(n)/A091137(n) unsigned.
Second row: A002206(n+1)/A002689(n) unsigned. See A141417(n) and A174727(n).
Third row: a(n)/A194506(n).

Alois P. Heinz, Table of n, a(n) for n = 0..200
Iaroslav V. Blagouchine, Three notes on Ser's and Hasse's representation for the zeta-functions, Integers (2018) 18A, Article #A3.

Cf. A194506 (denominator).

read("transforms3") ;
L := [seq(1/n,n=1..20)] ;
L1 := AKIYAMATANIGAWAi(L) ;
L2 := AKIYATANI(L1) ;
L3 := AKIYATANI(L2) ;
apply(numer,%) ; # R. J. Mathar, Aug 27 2011
# second Maple program:
b:= proc (n, k) option remember;
      `if`(n=0, 1/(k+1), b(n-1, k) -b(n-1, k+1)/n)
    end:
a:= n-> numer(b(n, 2)):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 27 2011

a[n_, 0] := 1/(n+1); a[n_, m_] := a[n, m] = a[n, m-1] - a[n+1, m-1]/m; Table[a[2, m], {m, 0, 21}] // Numerator (* Jean-François Alcover, Aug 09 2012 *)
Numerator@Table[(-1)^n (n + 1) Integrate[FunctionExpand[x Binomial[x, n + 1]], {x, 0, 1}], {n, 0, 20}] (* Vladimir Reshetnikov, Feb 01 2017 *)

a(n)/A194506(n) = (-1)^n * (n+1) * Integral_{0Vladimir Reshetnikov, Feb 01 2017

A174727 a(n) = A091137(n+1)/(n+1).

Original entry on oeis.org

2, 6, 8, 180, 288, 10080, 17280, 453600, 806400, 47900160, 87091200, 217945728000, 402361344000, 2241727488000, 4184557977600, 2000741783040000, 3766102179840000, 2838385676206080000, 5377993912811520000, 1686001091666411520000, 3211430650793164800000, 423033001181754163200000

Offset: 0

Paul Curtz, Mar 28 2010

nonn

Previous name: Inverse Akiyama-Tanigawa algorithm. From a column instead of a row. Bernoulli case A164555/A027642. We start from column 1, 1/2, 1/3, 1/4, 1/5 = A000012/A000027. First row: 1) (unreduced) 1, 1/2, 5/12, 9/24, 251/720 = A002657/A091137 (Cauchy from Bernoulli) (*); 2) (reduced) 1, 1/2, 5/12, 3/8, 251/720 = A002208/A002209 (Stirling and Bernoulli). Unreduced second row: 1/2, 1/6, 1/8, 19/180, 27/288, 863/10080 = A141417(n+1)/a(n).

(*) Reference page 56 (first row) and page 36 (upper main diagonal). From J. C. Adams (and Bashforth) numerical integration. See A165313 and A147998. See A002206 logarithm numbers (Gregory).

P. Curtz, Intégration numérique des systèmes différentiels .. . Note 12, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969.

Cf. A002689, A091137.

A091137[n_] := A091137[n] = Product[d, {d, Select[ Divisors[n] + 1, PrimeQ]}]*A091137[n-1]; A091137[0] = 1; a[n_] := A091137[n+1]/(n+1); Table[a[n], {n, 0, 18}] (* Jean-François Alcover_, Aug 14 2012 *)

f(n) = my(r =1); forprime(p=2, n+1, r*=p^(n\(p-1))); r; \\ A091137
a(n) = f(n+1)/(n+1); \\ Michel Marcus, Jun 30 2019

a(n) = A091137(n+1)/(n+1).

Extended up to a(18) by Jean-François Alcover, Aug 14 2012

New name and more terms from Michel Marcus, Jun 30 2019

A157982 Triangle T(n,m) read by rows which contains the coefficients [x^m] of the polynomial generating the numerators of the column A140825(.,n).

Original entry on oeis.org

1, 1, 2, -1, 0, 6, 1, 0, -6, 4, -19, 0, 120, -120, 30, 27

Offset: 0

Paul Curtz, Mar 10 2009

The first five polynomials describing the first five columns of A140825 are in A000012, A005408, A140811, A141530 and A157411.

			1;
1,2;     # 2n+1
-1,0,6;  # 6n^2-1
1,0,-6,4;  # 4n^3-6n^2+1, A141530
-19,0,120,-120,30;  # 30n^4-120n^3+120n^2-19, A157411

P Curtz Integration numerique des systemes differentiels a conditions initiales, C.C.S.A., Arcueil, 1969, p.36.

Cf. A141417.

A232853 Repeat n+1 times A091137(n).

Original entry on oeis.org

1, 2, 2, 12, 12, 12, 24, 24, 24, 24, 720, 720, 720, 720, 720, 1440, 1440, 1440, 1440, 1440, 1440, 60480, 60480, 60480, 60480, 60480, 60480, 60480, 120960, 120960, 120960, 120960, 120960, 120960, 120960, 120960, 3628800

Offset: 0

Paul Curtz, Dec 01 2013

A002657(n) and A091137(n) are linked to the Bernoulli numbers B n.
Unreduced differences table of A002657(n)/A091137(n):
1,            1/2,     5/12,    9/24,  251/720, 475/1440,...
-1/2,       -1/12,    -1/24, -19/720, -27/1440,... =-A141417(n+1)/A091137(n+1),
5/12,        1/24,   11/720, 11/1440,...
-9/24,    -19/720, -11/1440,...
251/720,  27/1440,...
-475/1440,... etc.
This is an autosequence of the second kind: its inverse binomial transform is the signed sequence with the main diagonal double of the first upper diagonal.
a(n) is the denominators written by antidiagonals.

			1,
2,   2,
12, 12, 12,
24, 24, 24, 24, etc.

P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales. Centre de Calcul Scientifique de l'Armement, Arcueil 1969. Pages 36 and 56.

Cf. A195287, A002208/A002209 (reduced autosequence), A193546, A174727, A165313.

Repeat n+1 times A091137(n). Triangle.

cp's OEIS Frontend

A027760 Denominator of Sum_{p prime, p-1 divides n} 1/p.

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A140811 a(n) = 6*n^2 - 1.

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A195189 Denominators of a sequence leading to gamma = A001620.

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A141530 a(n) = 4*n^3 - 6*n^2 + 1.

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A007525 Decimal expansion of log_2 e.

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A165313 Triangle T(n,k) = A091137(k-1) read by rows.

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A141530 a(n) = 4n^3 - 6n^2 + 1.