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A332786
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a(n) = numerator(-1/n + Sum_{k=1..n} 2^(k-1)/k).
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1
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0, 3, 3, 61, 25, 137, 343, 32663, 2357, 74689, 66671, 5299069, 2416531, 115545821, 106974277, 637525199, 74575583, 1588674349, 4496071973, 3234136824109, 1535024393629, 5843920343363, 5575228585159, 1961561381531581, 93953561866435, 9016382638527647, 2888981280567587, 200248741591132607, 96525489421136333
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OFFSET
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1,2
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COMMENTS
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If p > 3 is a prime, then p^2 | a(p).
Does the above statement follow from Wolstenholme's theorem?
If p is a Wolstenholme prime (A088164), then p^3 | a(p).
However, it should be noted that also 7^3 | a(7).
Conjecture: there are no pseudoprimes m such that m^2 | a(m).
Is 7^2 the only weak pseudoprime (i.e., a composite m such that m | a(m))?
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LINKS
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FORMULA
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a(n) = numerator(-2/n + S(n))/2 for odd n and a(n) = numerator(-2/n + S(n)) for even n, where S(n) = Sum_{k=1..n} 2^k/k, see A108866 / A229726.
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EXAMPLE
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a(5) = numerator(-1/5 + 1/1+2/2+4/3+8/4+16/5) = numerator(128/15 - 1/5) = numerator(25/3) = 25.
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MATHEMATICA
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n = 30; Numerator[Accumulate @ Table[(2^(k-1))/k, {k, 1, n}] - 1/Range[n]] (* Amiram Eldar, Feb 24 2020 *)
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PROG
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(PARI) a(n) = numerator(-1/n + sum(k=1, n, 2^(k-1)/k)); \\ Michel Marcus, Feb 24 2020
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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