|
|
A000146
|
|
From von Staudt-Clausen representation of Bernoulli numbers: a(n) = Bernoulli(2n) + Sum_{(p-1)|2n} 1/p.
(Formerly M1717 N0680)
|
|
10
|
|
|
1, 1, 1, 1, 1, 1, 2, -6, 56, -528, 6193, -86579, 1425518, -27298230, 601580875, -15116315766, 429614643062, -13711655205087, 488332318973594, -19296579341940067, 841693047573682616, -40338071854059455412, 2115074863808199160561, -120866265222965259346026
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,7
|
|
COMMENTS
|
The von Staudt-Clausen theorem states that this number is always an integer.
|
|
REFERENCES
|
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Th. 118.
Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, pp. 168-170.
H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Section 5.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
LINKS
|
|
|
MAPLE
|
A000146 := proc(n) local a , i, p; a := bernoulli(2*n) ; for i from 1 do p := ithprime(i) ; if (2*n) mod (p-1) = 0 then a := a+1/p ; elif p-1 > 2*n then break; end if; end do: a ; end proc: # R. J. Mathar, Jul 08 2011
|
|
MATHEMATICA
|
Table[ BernoulliB[2 n] + Total[ 1/Select[ Prime /@ Range[n+1], Divisible[2n, #-1] &]], {n, 1, 22}] (* Jean-François Alcover, Oct 12 2011 *)
|
|
PROG
|
(PARI) a(n)=if(n<1, 0, sumdiv(2*n, d, isprime(d+1)/(d+1))+bernfrac(2*n))
(Python)
from fractions import Fraction
from sympy import bernoulli, divisors, isprime
def A000146(n): return int(bernoulli(m:=n<<1)+sum(Fraction(1, d+1) for d in divisors(m, generator=True) if isprime(d+1))) # Chai Wah Wu, Apr 14 2023
|
|
CROSSREFS
|
|
|
KEYWORD
|
sign,nice,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
Signs courtesy of Antreas P. Hatzipolakis (xpolakis(AT)hol.gr)
|
|
STATUS
|
approved
|
|
|
|