A001067 -id:A001067 - cp's OEIS frontend

A262386 Numerators of a semi-convergent series leading to the third Stieltjes constant gamma_3.

0, 1, -17, 967, -4523, 33735311, -9301169, 127021899032857, -3546529522734769, 5633317707758173, -1935081812850766373, 779950247074296817622891, -1261508681536108282229, 350992098387568751020053498509, -17302487974885784968377519342317, 26213945071317075538702463006927083

Offset: 1

Iaroslav V. Blagouchine, Sep 20 2015

gamma_3 = + 1/120 - 17/1008 + 967/28800 - 4523/49896 + 33735311/101088000 - ..., see formulas (46)-(47) in the reference below.

			Numerators of -0/1, 1/120, -17/1008, 967/28800, -4523/49896, 33735311/101088000, ...

G. C. Greubel, Table of n, a(n) for n = 1..225
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, arXiv:1501.00740 [math.NT], 2015.

The sequence of denominators is A262387.

Cf. A001067, A001620, A002206, A006953, A075266, A082633, A086279, A086280, A195189, A262235, A262382, A262383, A262384, A262385.

a[n_] := Numerator[-BernoulliB[2*n]*(HarmonicNumber[2*n - 1]^3 - 3*HarmonicNumber[2*n - 1]*HarmonicNumber[2*n - 1, 2] + 2*HarmonicNumber[2*n - 1, 3])/(2*n)]; Table[a[n], {n, 1, 20}]

a(n) = numerator(-bernfrac(2*n)*(sum(k=1,2*n-1,1/k)^3 -3*sum(k=1,2*n-1,1/k)*sum(k=1,2*n-1,1/k^2) + 2*sum(k=1,2*n-1,1/k^3))/(2*n));

a(n) = numerator(-B_{2n}*(H^3_{2n-1}-3*H_{2n-1}*H^(2){2n-1}+2*H^(3){2n-1})/(2n)), where B_n, H_n and H^(k)_n are Bernoulli, harmonic and generalized harmonic numbers respectively.

A255505 Numerator of Bernoulli(2n)/(2n!).

Original entry on oeis.org

1, 1, -1, 1, -1, 1, -691, 1, -3617, 43867, -174611, 77683, -236364091, 657931, -3392780147, 1723168255201, -7709321041217, 151628697551, -26315271553053477373, 154210205991661, -261082718496449122051, 1520097643918070802691

Offset: 0

Jean-François Alcover, Feb 24 2015

This sequence is different from A001067 or A046968 or A141590, at least at a(52).

			The sequence Bernoulli(2n)/(2n!) (n >= 0) begins 1/2, 1/12, -1/120, 1/504, -1/1440, 1/3168, -691/3931200, 1/8640, -3617/41126400, ...

MathOverflow, What can be said about a function given its asymptotic expansion?

Cf. A000367, A001067, A046968, A141590, A255506 (denominator).

[Numerator(Bernoulli(2*n)/(2*Factorial(n))):n in [0..30]]; // Vincenzo Librandi, Feb 24 2015

Table[Numerator[BernoulliB[2 n]/(2 n!)], {n, 0, 25}]

a(n) = numerator(bernfrac(2*n)/(2*n!)); \\ Michel Marcus, Feb 24 2015

[numerator(bernoulli(2*n)/(2*factorial(n))) for n in (0..25)] # Bruno Berselli, Feb 24 2015

A262856 Numerators of the Nielsen-Jacobsthal series leading to Euler's constant.

Original entry on oeis.org

1, 43, 20431, 2150797323119, 9020112358835722225404403, 51551916515442115079024221439308876243677598340510141

Offset: 1

Iaroslav V. Blagouchine, Oct 03 2015

gamma = 1 - 1/12 - 43/420 - 20431/240240 - 2150797323119/36100888223400 - ..., see formula (36) in the reference below.

			Numerators of 1/12, 43/420, 20431/240240, 2150797323119/36100888223400, ...

G. C. Greubel, Table of n, a(n) for n = 1..10
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, arXiv:1501.00740 [math.NT], 2015.

Cf. A075266, A075267, A001620, A195189, A002657, A002790, A262235, A075266, A006953, A001067, A262858 (denominators of this series).

List(List([1..6],n->n*Sum([2^n+1..2^(n+1)],k->(-1)^(k+1)/k)),NumeratorRat); # Muniru A Asiru, Oct 29 2018

[Numerator(n*(&+[(-1)^(k+1)/k: k in [2^n+1..2^(n+1)]])): n in [1..6]]; // G. C. Greubel, Oct 28 2018

a[n_] := Numerator[n*Sum[(-1)^(k + 1)/k, {k, 2^n + 1, 2^(n + 1)}]]; Table[a[n], {n, 1, 8}]

a(n) = numerator(n*sum(k=2^n + 1,2^(n + 1),(-1)^(k + 1)/k));

a(n) = n * Sum_{k = 2^n + 1 .. 2^(n + 1)} (-1)^(k + 1)/k.

A262858 Denominators of the Nielsen-Jacobsthal series leading to Euler's constant.

Original entry on oeis.org

12, 420, 240240, 36100888223400, 236453376820564453502272320, 2225626015166235263233958200740039423756478781341512000

Offset: 1

Iaroslav V. Blagouchine, Oct 03 2015

gamma = 1 - 1/12 - 43/420 - 20431/240240 - 2150797323119/36100888223400 - ..., see formula (36) in the reference below.

			Denominators of 1/12, 43/420, 20431/240240, 2150797323119/36100888223400, ...

G. C. Greubel, Table of n, a(n) for n = 1..10
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, arXiv:1501.00740 [math.NT], 2015.

Cf. A075266, A075267, A001620, A195189, A002657, A002790, A262235, A075266, A006953, A001067, A262856 (numerators of this series).

List(List([1..6],n->n*Sum([2^n+1..2^(n+1)],k->(-1)^(k+1)/k)),DenominatorRat); # Muniru A Asiru, Oct 29 2018

[Denominator(n*(&+[(-1)^(k+1)/k: k in [2^n+1..2^(n+1)]])): n in [1..6]];  // G. C. Greubel, Oct 28 2018

a[n_] := Denominator[n*Sum[(-1)^(k + 1)/k, {k, 2^n + 1, 2^(n + 1)}]]; Table[a[n], {n, 1, 8}]

a(n) = denominator(n*sum(k=2^n + 1,2^(n + 1),(-1)^(k + 1)/k));

a(n) = n * Sum_{k = 2^n + 1 .. 2^(n + 1)} (-1)^(k + 1)/k.

A119766 Numbers n such that numerator of Bernoulli(n)/n is (apart from sign) 1 or a prime.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 26, 34, 36, 38, 42, 74, 114, 118, 396, 674, 1870, 4306, 22808

Offset: 1

N. J. A. Sloane, Jun 19 2006

nonn

In 1911 Ramanujan believed that the numerator of Bernoulli(n)/n for n even was (apart from sign) always either 1 or a prime. This is false.

			As an example, Bernoulli(20)/20 = -174611/6600, but 174611 = 283*617. - _Robert G. Wilson v_, Jun 22 2006

S. Ramanujan, Some properties of Bernoulli's numbers, J. Indian Math. Soc., 3 (1911), 219-234.

K. Ono, Honoring a gift from Kumbakonam, Notices Amer. Math. Soc., 53 (2006), 640-651.

Cf. A112548, A001067.

A119766 := proc(nmax) local numr; for n from 2 to nmax by 2 do numr := abs(numer(bernoulli(n)/n)) ; if numr = 1 or isprime(numr) then print(n) ; fi ; od ; end : A119766(2000) ; # R. J. Mathar, Jun 21 2006

OldPrimeQ[n_] := Abs[n]==1 || PrimeQ[Abs[n]]; Select[Range[2000], OldPrimeQ[Numerator[BernoulliB[ # ]/# ]] &] (* T. D. Noe, Jun 20 2006 *)

a(21) and a(22) from T. D. Noe, Jun 20 2006

A231273 Numerator of zeta(4n)/(zeta(2n) * Pi^(2n)).

Original entry on oeis.org

1, 1, 1, 691, 3617, 174611, 236364091, 3392780147, 7709321041217, 26315271553053477373, 261082718496449122051, 2530297234481911294093, 5609403368997817686249127547, 61628132164268458257532691681, 354198989901889536240773677094747

Offset: 0

Leo Depuydt, Nov 07 2013

Integer component of the numerator of a close variant of Euler's infinite prime product zeta(2n) = Product_{k>=1} (prime(k)^(2n))/(prime(k)^(2n)-1), namely with all minus signs changed into plus signs, as follows: zeta(4n)/zeta(2n) = Product_{k>=1} (prime(k)^(2n))/(prime(k)^(2n)+1). The transcendental component is Pi^(2n).
For a detailed account of the results, including proof and relation to the zeta function, see Links for the PDF file submitted as supporting material.
The reference to Apostol is to a discussion of the equivalence of 1) zeta(2s)/zeta(s) and 2) a related infinite prime product, that is, Product_{sigma>1} prime(n)^s/(prime(n)^s + 1), with s being a complex variable such that s = sigma + i*t where sigma and t are real (following Riemann), using a type of proof different from the one posted below involving zeta(4n)/zeta(2n). On this, see also Hardy and Wright cited below. - Leo Depuydt, Nov 22 2013, Nov 27 2013
The background of the sequence is now described in the link below to L. Depuydt, The Prime Sequence ... . - Leo Depuydt, Aug 22 2014
From Robert Israel, Aug 22 2014: (Start)
Numerator of (-1)^n*B(4*n)*4^n*(2*n)!/(B(2*n)*(4*n)!), where B(n) are the Bernoulli numbers (see A027641 and A027642).
Not the same as abs(A001067(2*n)): they differ first at n=17.
(End)

T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1976, p. 231.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fourth Edition, Clarendon Press, 1960, p. 255.

Leo Depuydt, Details, including proof and relation to zeta function

Cf. A231327 (corresponding denominator).
Cf. A114362 and A114363 (closely related results).
Cf. A001067, A046968, A046988, A098087, A141590, A156036 (same number sequence, though in various transformations (alternation of signs, intervening numbers, and so on)).
Cf. A027641 and A027642

seq(numer((-1)^n*bernoulli(4*n)*4^n*(2*n)!/(bernoulli(2*n)*(4*n)!)),n=0..100); # Robert Israel, Aug 22 2014

Numerator[Table[Zeta[4n]/(Zeta[2n] * Pi^(2n)), {n, 0, 15}]] (* T. D. Noe, Nov 18 2013 *)

A358625 a(n) = numerator of Bernoulli(n, 1) / n for n >= 1, a(0) = 1.

Original entry on oeis.org

1, 1, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -691, 0, 1, 0, -3617, 0, 43867, 0, -174611, 0, 77683, 0, -236364091, 0, 657931, 0, -3392780147, 0, 1723168255201, 0, -7709321041217, 0, 151628697551, 0, -26315271553053477373, 0, 154210205991661, 0, -261082718496449122051

Offset: 0

Peter Luschny, Dec 02 2022

The rational numbers r(n) = Bernoulli(n, 1) / n are called the 'divided Bernoulli numbers'. r(n) is a p-integer for all primes p if p - 1 does not divide n. This is sometimes called 'Adams's theorem' (Ireland and Rosen). The important Kummer congruences for the Bernoulli numbers (1851) are stated in terms of the r(n).

			Rationals: 1, 1/2, 1/12, 0, -1/120, 0, 1/252, 0, -1/240, 0, 1/132, ...
Note that a(68) = -4633713579924631067171126424027918014373353 but A120082(68) = -125235502160125163977598011460214000388469.

Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, Vol. 84, Graduate Texts in Mathematics, Springer-Verlag, 2nd edition, 1990. [Prop. 15.2.4, p. 238]

Peter Luschny, Table of n, a(n) for n = 0..300
Arnold Adelberg, Shaofang Hong and Wenli Ren, Bounds of Divided Universal Bernoulli Numbers and Universal Kummer Congruences, Proceedings of the American Mathematical Society, Vol. 136(1), 2008, p. 61-71.
Bernd C. Kellner, The structure of Bernoulli numbers, arXiv:math/0411498 [math.NT], 2004.

Cf. A001067, A342318, A120080, A120082, A120084, A120086, A079612, A075180, A060054.
Cf. A164555 / A027642.
For the denominators cf. A006953, A036283, A053657, A202318.

Concatenation([1, 1], List([2..45], n-> NumeratorRat(Bernoulli(n)/(n)))); # G. C. Greubel, Sep 19 2019

[1, 1] cat [Numerator(Bernoulli(n)/(n)): n in [2..45]]; // G. C. Greubel, Sep 19 2019

A358625 := n -> ifelse(n = 0, 1, numer(bernoulli(n, 1) / n)):
seq(A358625(n), n = 0.. 40);
# Alternative:
egf := 1 + x + log(1 - exp(-x)) - log(x): ser := series(egf, x, 42):
seq(numer(n! * coeff(ser, x, n)), n = 0..40);

Join[{1, 1}, Table[Numerator[BernoulliB[n] / n], {n, 2, 45}]]

a(n) = if (n<=1, 1, numerator(bernfrac(n)/n)); \\ Michel Marcus, Feb 24 2015

a(n) = numerator(n! * [x^n](1 + x + log(1 - exp(-x)) - log(x))).
a(n) = numerator(-zeta(1 - n)) for n >= 1.
a(n) = numerator(Euler(n-1, 1) / (2*(2^n - 1))) for n >= 1.
denominator(r(2*n)) = A006953(n) for n >= 1.
denominator(r(2*n)) / 2 = A036283(n) for n >= 1.
denominator(r(2*n)) / 12 = A202318(n) for n >= 1.
denominator(r(2*n)) = (1/2) * A053657(2*n+1) / A053657(2*n-1) for n >= 1.

A043303 Numerator of B(4n+2)/(2n+1) where B(m) are the Bernoulli numbers.

Original entry on oeis.org

1, 1, 1, 1, 43867, 77683, 657931, 1723168255201, 151628697551, 154210205991661, 1520097643918070802691, 25932657025822267968607, 19802288209643185928499101, 29149963634884862421418123812691, 2913228046513104891794716413587449, 396793078518930920708162576045270521

Offset: 0

Benoit Cloitre, Apr 04 2002

Note that numerator of B(2n)/n is odd so B(2n)/(2n), B(2n)/(4n), etc. have the same numerators. - Michael Somos, Feb 01 2004

Bruce Berndt, Ramanujan's Notebooks Part II, Springer-Verlag; see Infinite series, p. 262.

Seiichi Manyama, Table of n, a(n) for n = 0..156

Cf. A001067, A043304.

seq(numer(bernoulli(4*n+2)/(2*n+1)),n=0..30); # Robert Israel, Sep 18 2016

Table[BernoulliB[4n+2]/(2n+1),{n,0,20}]//Numerator (* Harvey P. Dale, Aug 13 2018 *)

a(n)=if(n<0,0,numerator(bernfrac(4*n+2)/(2*n+1)))

B(4*n+2)/(8*n+4) = Sum_{k>=1} k^(4*n+1)/(exp(2*Pi*k)-1).

a(n) = A001067(2n+1).

A231327 Denominator of rational component of zeta(4n)/zeta(2n).

Original entry on oeis.org

1, 15, 105, 675675, 34459425, 16368226875, 218517792968475, 30951416768146875, 694097901592400930625, 23383376494609715287281703125, 2289686345687357378035370971875, 219012470258383844016431785453125, 4791965046290912124048163518904807546875

Offset: 0

Leo Depuydt, Nov 07 2013

Denominator of a close variant of Euler's infinite prime product zeta(2n) = Product_{k>=1} (prime(k)^(2n))/(prime(k)^(2n)-1), namely with all minus signs changed into plus signs, as follows: zeta(4n)/zeta(2n) = Product_{k>=1} prime(k)^(2n)/(prime(k)^(2n)+1).
For a detailed account of the results in question, including proof and relation to the zeta function, see the PDF file submitted as supporting material in A231273.
The reference to Apostol below is a discussion of the equivalence of 1) zeta(2s)/zeta(s) and 2) a related infinite prime product, that is, Product_{sigma>1} prime(n)^s/(prime(n)^s + 1), with s being a complex variable such that s = sigma + i*t where sigma and t are real (following Riemann), using a type of proof different from the one posted below involving zeta(4n)/zeta(2n). - Leo Depuydt, Nov 22 2013
Denominator of B(4*n)*4^n*(2*n)!/(B(2*n)*(4*n)!) where B(n) are the Bernoulli numbers (see A027641 and A027642). - Robert Israel, Aug 22 2014

T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1976, p. 231.

Cf. A231273 (the corresponding numerator).
Cf. A114362 and A114363 (closely related results).
Cf. A001067, A046968, A046988, A098087, A141590, and A156036 (same number sequence as found in numerator, though in various transformations (alternation of sign, intervening numbers, and so on)).
Cf. A027641 and A027642.

seq(denom(bernoulli(4*n)*4^n*(2*n)!/(bernoulli(2*n)*(4*n)!)),n=0..100); # Robert Israel, Aug 22 2014

Denominator[Table[Zeta[4 n]/Zeta[2 n], {n, 0, 15}]] (* T. D. Noe, Nov 15 2013 *)

A342318 a(n) = numerator(((i^n * PolyLog(1 - n, -i) + (-i)^n * PolyLog(1 - n, i))) / (4^n - 2^n)) if n > 0 and a(0) = 1. Here i denotes the imaginary unit.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 61, 1, 1385, 1, 50521, 691, 2702765, 1, 199360981, 3617, 19391512145, 43867, 2404879675441, 174611, 370371188237525, 77683, 69348874393137901, 236364091, 15514534163557086905, 657931, 4087072509293123892361, 3392780147, 1252259641403629865468285

Offset: 0

Peter Luschny, Mar 22 2021

The defining formula simultaneously represents the numerators of the unsigned divided Bernoulli numbers and the unsigned Euler secant numbers. Some authors consider the divided Bernoulli numbers B(n)/n to be more fundamental than B(n). For instance, B(n)/n is a p-integer for all primes p for which p - 1 does not divide n (see Ireland and Rosen).

			r(n) = 1, 1/2, 1/12, 1/56, 1/120, 5/992, 1/252, 61/16256, 1/240, 1385/261632, 1/132, ...

K. Ireland and M. Rosen, A classical introduction to modern number theory, vol. 84, Graduate Texts in Mathematics. Springer-Verlag, 2nd edition, 1990. [Prop. 15.2.4, p. 238]

Arnold Adelberg, Shaofang Hong and Wenli Ren, Bounds of Divided Universal Bernoulli Numbers and Universal Kummer Congruences, Proceedings of the American Mathematical Society, Vol. 136(1), 2008, p. 61-71.
Bernd C. Kellner, The structure of Bernoulli numbers, arXiv:math/0411498 [math.NT], 2004.

Cf. A342319 (denominator), A001067, A000364, A122045.

a := n -> `if`(n <= 2, 1, `if`(n::even, numer(abs(bernoulli(n))/n), abs(euler(n - 1)))); seq(a(n), n = 0..29);

r[s_] := If[s == 0, 1, (I^s PolyLog[1 - s, -I] + (-I)^s PolyLog[1 - s, I]) / (4^s - 2^s)]; Table[r[n], {n, 0, 29}] // Numerator

a(2*n) = |A001067(n)| for n > 0.

a(2*n+1) = A000364(n) = |A122045(2*n)|.

cp's OEIS Frontend

A262386 Numerators of a semi-convergent series leading to the third Stieltjes constant gamma_3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A255505 Numerator of Bernoulli(2n)/(2n!).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

A262856 Numerators of the Nielsen-Jacobsthal series leading to Euler's constant.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Formula

A262858 Denominators of the Nielsen-Jacobsthal series leading to Euler's constant.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Formula

A119766 Numbers n such that numerator of Bernoulli(n)/n is (apart from sign) 1 or a prime.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A231273 Numerator of zeta(4n)/(zeta(2n) * Pi^(2n)).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

A358625 a(n) = numerator of Bernoulli(n, 1) / n for n >= 1, a(0) = 1.

Views