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A330718
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a(n) = numerator(Sum_{k=1..n} (2^k-2)/k).
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6
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0, 1, 3, 13, 25, 137, 245, 871, 517, 4629, 8349, 45517, 83317, 1074679, 1992127, 7424789, 13901189, 78403447, 147940327, 280060651, 531718651, 11133725681, 21243819521, 40621501691, 15565330735, 388375065019, 248882304985, 479199924517, 923951191477, 2973006070891
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OFFSET
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1,3
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COMMENTS
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If p > 3 is prime, then p^2 | a(p).
Note the similarity to Wolstenholme's theorem.
Conjecture: for n > 3, if n^2 | a(n), then n is prime.
Are there the weak pseudoprimes m such that m | a(m)?
Primes p such that p^3 | a(p) are probably A088164.
If p is an odd prime, then a(p+1) == A330719(p+1) (mod p).
If p > 3 is a prime, then p^2 | numerator(Sum_{k=1..p+1} F(k)), where F(n) = Sum_{k=1..n} (2^(k-1)-1)/k. Cf. A027612 (a weaker divisibility).
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LINKS
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FORMULA
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a(n) = numerator(Sum_{k=1..n} (2^(k-1)-1)/k).
a(n) = numerator(-(2^(n+1)*LerchPhi(2,1,n+1) + Pi*i + 2*HarmonicNumber(n))). - G. C. Greubel, Dec 28 2019
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MAPLE
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seq(numer(add((2^k -2)/k, k = 1..n)), n = 1..30); # G. C. Greubel, Dec 28 2019
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MATHEMATICA
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Numerator @ Accumulate @ Array[(2^# - 2)/# &, 30]
Table[Numerator[Simplify[-(2^(n+1)*LerchPhi[2, 1, n+1] +Pi*I +2*HarmonicNumber[n])]], {n, 30}] (* G. C. Greubel, Dec 28 2019 *)
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PROG
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(PARI) a(n) = numerator(sum(k=1, n, (2^k-2)/k)); \\ Michel Marcus, Dec 28 2019
(Magma) [Numerator( &+[(2^k -2)/k: k in [1..n]] ): n in [1..30]]; // G. C. Greubel, Dec 28 2019
(Sage) [numerator( sum((2^k -2)/k for k in (1..n)) ) for n in (1..30)] # G. C. Greubel, Dec 28 2019
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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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STATUS
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approved
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