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A000146 From von Staudt-Clausen representation of Bernoulli numbers: a(n) = Bernoulli(2n) + Sum_{(p-1)|2n} 1/p.
(Formerly M1717 N0680)
6
1, 1, 1, 1, 1, 1, 2, -6, 56, -528, 6193, -86579, 1425518, -27298230, 601580875, -15116315766, 429614643062, -13711655205087, 488332318973594, -19296579341940067, 841693047573682616, -40338071854059455412 (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.

Knuth, D. E.; Buckholtz, Thomas J. 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.

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

T. D. Noe, Table of n, a(n) for n=1..100

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

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-Fran├žois Alcover, Oct 12 2011 *)

PROG

(PARI) a(n)=if(n<1, 0, sumdiv(2*n, d, isprime(d+1)/(d+1))+bernfrac(2*n))

CROSSREFS

Cf. also A002882, A003245, A127187, A127188.

Sequence in context: A181509 A213026 A074023 * A211933 A167010 A014070

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

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Last modified July 29 01:32 EDT 2016. Contains 275161 sequences.