A036283 -id:A036283 - cp's OEIS frontend

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

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.

A228838 a(n) = n * A002445(n).

Original entry on oeis.org

0, 6, 60, 126, 120, 330, 16380, 42, 4080, 7182, 3300, 1518, 32760, 78, 12180, 214830, 8160, 102, 34545420, 114, 270600, 37926, 15180, 6486, 1113840, 1650, 41340, 21546, 24360, 10266, 1703601900, 186, 16320, 2135826, 1020, 164010, 5043631320, 222, 1140

Offset: 0

Paul Curtz, Sep 05 2013

nonn

a(n+1) is a multiple of A040031(n+1), sequence of period 2: 6, 12.
a(n) is divisible by A040879(n)=30 followed by the sequence of period 2: 6, 60. See A040214 and A165734.
Note that A164877(n) + A000367(n) = A164558(2n).

			a(0)=0*1, a(1)=1*6, a(2)=2*30=60,, a(3)=3*42=126.

Colin Barker, Table of n, a(n) for n = 0..1000

PARI
```
a(n)=n*denominator(bernfrac(2*n))
```

a(n) = A176328(2n) - A000367(n).
a(n) = A164877(n)/2.
a(n+1) = A111008(n) * A036283(n+1).
2*a(n) = A164558(2n) - A000367(n).
a(n) = A164558(2n) - A176328(2n).

Typo in data fixed by Colin Barker, Jul 03 2015

cp's OEIS Frontend

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

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