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A330718 a(n) = numerator(Sum_{k=1..n} (2^k-2)/k).

%I A330718 #72 Mar 30 2025 10:55:41
%S A330718 0,1,3,13,25,137,245,871,517,4629,8349,45517,83317,1074679,1992127,
%T A330718 7424789,13901189,78403447,147940327,280060651,531718651,11133725681,
%U A330718 21243819521,40621501691,15565330735,388375065019,248882304985,479199924517,923951191477,2973006070891
%N A330718 a(n) = numerator(Sum_{k=1..n} (2^k-2)/k).
%C A330718 If p > 3 is prime, then p^2 | a(p).
%C A330718 Note the similarity to Wolstenholme's theorem.
%C A330718 Conjecture: for n > 3, if n^2 | a(n), then n is prime.
%C A330718 Are there the weak pseudoprimes m such that m | a(m)?
%C A330718 Primes p such that p^3 | a(p) are probably A088164.
%C A330718 If p is an odd prime, then a(p+1) == A330719(p+1) (mod p).
%C A330718 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).
%H A330718 G. C. Greubel, <a href="/A330718/b330718.txt">Table of n, a(n) for n = 1..1000</a>
%H A330718 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WolstenholmesTheorem.html">Wolstenholme's Theorem</a>
%F A330718 a(n) = numerator(Sum_{k=1..n} (2^(k-1)-1)/k).
%F A330718 a(n+1) = numerator(a(n)/A330719(n) + A225101(n+1)/(2*A159353(n+1))).
%F A330718 a(p) = a(p-1) + A007663(n)*A330719(p-1) for p = prime(n) > 2.
%F A330718 a(n) = numerator(-(2^(n+1)*LerchPhi(2,1,n+1) + Pi*i + 2*HarmonicNumber(n))). - _G. C. Greubel_, Dec 28 2019
%F A330718 a(n) = numerator(A279683(n)/n!) for n > 0. - _Amiram Eldar_ and _Thomas Ordowski_, Jan 15 2020
%F A330718 For n > 1, a(n) = A000265(A290347(n)). - _Thomas Ordowski_, Mar 29 2025
%e A330718 Numerators of 0, 1, 3, 13/2, 25/2, 137/6, 245/6, ...
%p A330718 seq(numer(add((2^k -2)/k, k = 1..n)), n = 1..30); # _G. C. Greubel_, Dec 28 2019
%t A330718 Numerator @ Accumulate @ Array[(2^# - 2)/# &, 30]
%t A330718 Table[Numerator[Simplify[-(2^(n+1)*LerchPhi[2,1,n+1] +Pi*I +2*HarmonicNumber[n])]], {n,30}] (* _G. C. Greubel_, Dec 28 2019 *)
%o A330718 (PARI) a(n) = numerator(sum(k=1, n, (2^k-2)/k)); \\ _Michel Marcus_, Dec 28 2019
%o A330718 (Magma) [Numerator( &+[(2^k -2)/k: k in [1..n]] ): n in [1..30]]; // _G. C. Greubel_, Dec 28 2019
%o A330718 (Sage) [numerator( sum((2^k -2)/k for k in (1..n)) ) for n in (1..30)] # _G. C. Greubel_, Dec 28 2019
%Y A330718 Cf. A000265, A001008, A007663, A088164, A159353, A225101, A279683, A290347, A330719.
%K A330718 nonn,frac
%O A330718 1,3
%A A330718 _Amiram Eldar_ and _Thomas Ordowski_, Dec 28 2019