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A000146
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From von Staudt-Clausen representation of Bernoulli numbers: a(n) = Bernoulli(2n) + Sum_{(p-1)|2n} 1/p.
(Formerly M1717 N0680)
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6
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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; internal format)
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OFFSET
| 1,7
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COMMENTS
| The von Staudt-Clausen theorem states that this number is always an integer.
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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.
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).
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..100
Eric Weisstein's World of Mathematics, von Staudt-Clausen Theorem
Index entries for sequences related to Bernoulli numbers.
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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
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MATHEMATICA
| Table[ BernoulliB[2 n] + Total[ 1/Select[ Prime /@ Range[n+1], Divisible[2n, #-1] &]], {n, 1, 22}] (* From Jean-François Alcover, Oct 12 2011 *)
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PROG
| (PARI) a(n)=if(n<1, 0, sumdiv(2*n, d, isprime(d+1)/(d+1))+bernfrac(2*n))
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CROSSREFS
| Cf. also A002882, A003245, A127187, A127188.
Sequence in context: A193473 A181509 A074023 * A167010 A014070 A198445
Adjacent sequences: A000143 A000144 A000145 * A000147 A000148 A000149
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KEYWORD
| sign,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Signs courtesy of xpolakis(AT)hol.gr (Antreas P. Hatzipolakis). More terms from Michael Somos
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