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

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

Author

N. J. A. Sloane

Keywords

• sign
• nice
• easy

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.

Crossrefs

Cf. also A002882, A003245, A127187, A127188.

Programs

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 *)

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

Extensions

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

More terms from Michael Somos