A000146 From von Staudt-Clausen representation of Bernoulli numbers: a(n) = Bernoulli(2n) + Sum_{(p-1)|2n} 1/p. (Formerly M1717 N0680)

1, 1, 1, 1, 1, 1, 2, -6, 56, -528, 6193, -86579, 1425518, -27298230, 601580875, -15116315766, 429614643062, -13711655205087, 488332318973594, -19296579341940067, 841693047573682616, -40338071854059455412, 2115074863808199160561, -120866265222965259346026

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

Seiichi Manyama, Table of n, a(n) for n = 1..317 (first 100 terms from T. D. Noe)

Joerg Arndt, Table of n, a(n) for n = 1..1000 (contains terms with more than 1000 decimal digits)

Daniel Hoyt, Python 3 program for A000146.

Donald E. Knuth and Thomas J. Buckholtz, Computation of tangent, Euler and Bernoulli numbers, Math. Comp. 21 1967 663-688. [Annotated scanned copy]

Donald E. Knuth and Thomas J. Buckholtz, Computation of tangent, Euler and Bernoulli numbers, Math. Comp. 21 1967 663-688.

R. Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.

Eric Weisstein's World of Mathematics, von Staudt-Clausen Theorem

Index entries for sequences related to Bernoulli numbers.

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

Cf. also A002882, A003245, A127187, A127188.

Sequence in context: A213026 A074023 A354315 * A318001 A211933 A167010

Adjacent sequences: A000143 A000144 A000145 * A000147 A000148 A000149

Keyword

sign,nice,easy

Author

N. J. A. Sloane

Extensions

Signs courtesy of Antreas P. Hatzipolakis (xpolakis(AT)hol.gr)

More terms from Michael Somos

Status

approved