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A216921
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a(n) = 2*n+1 - gpf(denominator(B°(2*n))) where B°(n) are Zagier's modification of the Bernoulli numbers and gpf(n) is the greatest prime factor of n.
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1
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0, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 2, 24, 0, 0, 2, 32, 0, 2, 0, 0, 2, 0, 2, 40, 0, 2, 28, 0, 0, 2, 34, 0, 2, 0, 0, 2, 40, 0, 2, 0, 2, 84, 0, 2, 46, 92, 0, 2, 0, 0, 2, 0, 0, 2, 0, 2, 58, 116, 60, 120, 64, 0, 2, 0, 2, 132, 0, 0, 2, 140, 72, 144, 0, 0, 2, 132, 0
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OFFSET
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1,4
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COMMENTS
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Dixit and others (see link, p.13) wrote: "The data suggests that the prime factors of alpha(2n) [= denominator(B°(2n))] are bounded by 2n + 1." If this is true a(n) will never become negative. The data also suggests that a(n) = 0 only if 2n+1 is prime.
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LINKS
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FORMULA
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MAPLE
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zb2 := denom(add(binomial(2*n+r, 2*r)*bernoulli(r)/(2*n+r), r=0..2*n));
F := ifactors(zb2)[2]; 2*n+1-F[nops(F)][1] end;
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MATHEMATICA
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b[n_] := Sum[Binomial[n + k, 2*k]*BernoulliB[k]/(n + k), {k, 0, n}] // Denominator;
a[n_] := 2*n + 1 - FactorInteger[b[2*n]][[-1, 1]];
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PROG
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(PARI) f(n) = denominator(sum(r=0, n, binomial(n+r, 2*r)*bernfrac(r)/(n+r))); \\ A216923
a(n) = 2*n+1 - vecmax(factor(f(2*n))[, 1]); \\ Michel Marcus, Sep 29 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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