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

Author

Peter Luschny, Dec 02 2022

Keywords

• sign
• frac

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).

Examples

			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.
		

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.

Crossrefs

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

Cf. A164555 / A027642.

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

Programs

GAP

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

Magma

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

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}]]

PARI

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

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.