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

%I A358625 #27 Dec 05 2022 08:51:35
%S A358625 1,1,1,0,-1,0,1,0,-1,0,1,0,-691,0,1,0,-3617,0,43867,0,-174611,0,77683,
%T A358625 0,-236364091,0,657931,0,-3392780147,0,1723168255201,0,-7709321041217,
%U A358625 0,151628697551,0,-26315271553053477373,0,154210205991661,0,-261082718496449122051
%N A358625 a(n) = numerator of Bernoulli(n, 1) / n for n >= 1, a(0) = 1.
%C A358625 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).
%D A358625 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]
%H A358625 Peter Luschny, <a href="/A358625/b358625.txt">Table of n, a(n) for n = 0..300</a>
%H A358625 Arnold Adelberg, Shaofang Hong and Wenli Ren, <a href="https://doi.org/10.1090/S0002-9939-07-09025-9">Bounds of Divided Universal Bernoulli Numbers and Universal Kummer Congruences</a>, Proceedings of the American Mathematical Society, Vol. 136(1), 2008, p. 61-71.
%H A358625 Bernd C. Kellner, <a href="http://arxiv.org/abs/math/0411498">The structure of Bernoulli numbers</a>, arXiv:math/0411498 [math.NT], 2004.
%F A358625 a(n) = numerator(n! * [x^n](1 + x + log(1 - exp(-x)) - log(x))).
%F A358625 a(n) = numerator(-zeta(1 - n)) for n >= 1.
%F A358625 a(n) = numerator(Euler(n-1, 1) / (2*(2^n - 1))) for n >= 1.
%F A358625 denominator(r(2*n)) = A006953(n) for n >= 1.
%F A358625 denominator(r(2*n)) / 2 = A036283(n) for n >= 1.
%F A358625 denominator(r(2*n)) / 12 = A202318(n) for n >= 1.
%F A358625 denominator(r(2*n)) = (1/2) * A053657(2*n+1) / A053657(2*n-1) for n >= 1.
%e A358625 Rationals: 1, 1/2, 1/12, 0, -1/120, 0, 1/252, 0, -1/240, 0, 1/132, ...
%e A358625 Note that a(68) = -4633713579924631067171126424027918014373353 but A120082(68) = -125235502160125163977598011460214000388469.
%p A358625 A358625 := n -> ifelse(n = 0, 1, numer(bernoulli(n, 1) / n)):
%p A358625 seq(A358625(n), n = 0.. 40);
%p A358625 # Alternative:
%p A358625 egf := 1 + x + log(1 - exp(-x)) - log(x): ser := series(egf, x, 42):
%p A358625 seq(numer(n! * coeff(ser, x, n)), n = 0..40);
%t A358625 Join[{1, 1}, Table[Numerator[BernoulliB[n] / n], {n, 2, 45}]]
%o A358625 (PARI) a(n) = if (n<=1, 1, numerator(bernfrac(n)/n)); \\ _Michel Marcus_, Feb 24 2015
%o A358625 (Magma) [1, 1] cat [Numerator(Bernoulli(n)/(n)): n in [2..45]]; // _G. C. Greubel_, Sep 19 2019
%o A358625 (GAP) Concatenation([1, 1], List([2..45], n-> NumeratorRat(Bernoulli(n)/(n)))); # _G. C. Greubel_, Sep 19 2019
%Y A358625 Cf. A001067, A342318, A120080, A120082, A120084, A120086, A079612, A075180, A060054.
%Y A358625 Cf. A164555 / A027642.
%Y A358625 For the denominators cf. A006953, A036283, A053657, A202318.
%K A358625 sign,frac
%O A358625 0,13
%A A358625 _Peter Luschny_, Dec 02 2022