A174341 a(n) = numerator(Bernoulli(n, 1) + 1/(n+1)).

2, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, -37, 1, 37, 1, -211, 1, 2311, 1, -407389, 1, 37153, 1, -1181819909, 1, 76977929, 1, -818946931, 1, 277930363757, 1, -84802531453217, 1, 90219075042851, 1, -711223555487930419, 1, 12696640293313423, 1, -6367871182840222481, 1, 35351107998094669831, 1, -83499808737903072705023, 1, 12690449182849194963361, 1

Offset

0,1

Comments

a(n) is numerator of (A164555(n)/A027642(n) + 1/(n+1)).

1/(n+1) and Bernoulli(n,1) are autosequences in the sense that they remain the same (up to sign) under inverse binomial transform. This feature is kept for their sum, a(n)/A174342(n) = 2, 1, 1/2, 1/4, 1/6, 1/6, 1/6, 1/8, 7/90, 1/10, ...

Similar autosequences are also A000045, A001045, A113405, A000975 preceded by two zeros, and A140096.

Conjecture: the numerator of (A164555(n)/(n+1) + A027642(n)/(n+1)^2) is a(n) and the denominator of this fraction is equal to 1 if and only if n+1 is prime or 1. Cf. A309132. - Thomas Ordowski, Jul 09 2019

The "if" part of the conjecture is true: see the theorems in A309132 and A326690. The values of the numerator when n+1 is prime are A327033. - Jonathan Sondow, Aug 15 2019

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

OEIS Wiki, Autosequence

Maple

B := proc(n) if n <> 1 then bernoulli(n) ; else -bernoulli(n) ; end if; end proc:
A174341 := proc(n) B(n)+1/(n+1) ; numer(%) ; end proc: # R. J. Mathar, Nov 19 2010

Mathematica

a[n_] := Numerator[BernoulliB[n, 1] + 1/(n + 1)];
Table[a[n], {n, 0, 47}] (* Peter Luschny, Jul 13 2019 *)

Prog

(PARI)
B(n)=if(n!=1, bernfrac(n), -bernfrac(n));
a(n)=numerator(B(n) + 1/(n + 1));
for(n=0, 50, print1(a(n), ", ")) \\ Indranil Ghosh, Jun 19 2017
(Python)
from sympy import bernoulli, Integer
def B(n): return bernoulli(n) if n != 1 else -bernoulli(n)
def a(n): return (B(n) + 1/Integer(n + 1)).numerator() # Indranil Ghosh, Jun 19 2017
(Magma) [2, 1] cat [Numerator(Bernoulli(n)+1/(n+1)): n in [2..40]]; // Vincenzo Librandi, Jul 18 2019

Crossrefs

Cf. A164555, A027642, A174342 (denominators), A025529, A003506, A309132, A326690, A327033.

Sequence in context: A290602 A255404 A078090 * A168516 A294335 A194321

Adjacent sequences: A174338 A174339 A174340 * A174342 A174343 A174344

Keyword

sign,frac

Author

Paul Curtz, Mar 16 2010

Extensions

Reformulation of the name by Peter Luschny, Jul 13 2019

Status

approved