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

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

0,13

Comments

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).

References

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]

Links

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.

Formula

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.

Example

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.

Maple

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);

Mathematica

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

Prog

(PARI) a(n) = if (n<=1, 1, numerator(bernfrac(n)/n)); \\ Michel Marcus, Feb 24 2015
(Magma) [1, 1] cat [Numerator(Bernoulli(n)/(n)): n in [2..45]]; // G. C. Greubel, Sep 19 2019
(GAP) Concatenation([1, 1], List([2..45], n-> NumeratorRat(Bernoulli(n)/(n)))); # G. C. Greubel, Sep 19 2019

Crossrefs

Cf. A001067, A342318, A120080, A120082, A120084, A120086, A079612, A075180, A060054.

Cf. A164555 / A027642.

For the denominators cf. A006953, A036283, A053657, A202318.

Sequence in context: A060054 A120082 A120084 * A249699 A141588 A281331

Adjacent sequences: A358622 A358623 A358624 * A358626 A358627 A358628

Keyword

sign,frac

Author

Peter Luschny, Dec 02 2022

Status

approved