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A330719 a(n) = denominator(Sum_{k=1..n} (2^(k-1) - 1)/k).

%I A330719 #43 Apr 04 2025 18:14:30
%S A330719 1,2,2,4,4,12,12,24,8,40,40,120,120,840,840,1680,1680,5040,5040,5040,
%T A330719 5040,55440,55440,55440,11088,144144,48048,48048,48048,80080,80080,
%U A330719 160160,160160,2722720,544544,4900896,4900896,93117024,93117024,465585120,465585120,465585120
%N A330719 a(n) = denominator(Sum_{k=1..n} (2^(k-1) - 1)/k).
%C A330719 Conjecture: if p is an odd prime, then p | A330718(p+1) - a(p+1).
%C A330719 Below 10^6 there is only one pseudoprime, namely 25. Are there others?
%C A330719 Primes p such that p^2 | A330718(p+1) - a(p+1) are 3, 5, 45827, ...
%H A330719 Metin Sariyar, <a href="/A330719/b330719.txt">Table of n, a(n) for n = 1..500</a>
%F A330719 a(n) = denominator(-(2^n*LerchPhi(2,1,n+1) + Pi*i/2 + HarmonicNumber(n))). - _G. C. Greubel_, Dec 28 2019
%F A330719 a(n) = denominator(A279683(n)/n!) for n > 0. - _Amiram Eldar_ and _Thomas Ordowski_, Jan 15 2020
%F A330719 A000265(a(n)) = A290348(n). - _Thomas Ordowski_, Mar 29 2025
%e A330719 Denominators of 0, 1/2, 3/2, 13/4, 25/4, 137/12, 245/12, ...
%p A330719 seq(denom(add((2^(k-1)-1)/k, k = 1..n)), n = 1..45); # _G. C. Greubel_, Dec 28 2019
%t A330719 Denominator@Accumulate@Array[(2^(#-1) -1)/# &, 45]
%t A330719 Table[Denominator[-(2^n*LerchPhi[2, 1, n+1] +Pi*I/2 +HarmonicNumber[n])], {n, 45}] (* _G. C. Greubel_, Dec 28 2019 *)
%o A330719 (PARI) a(n) = denominator(sum(k=1, n, (2^(k-1)-1)/k)); \\ _Michel Marcus_, Dec 28 2019
%o A330719 (Magma) [Denominator( &+[(2^(k-1)-1)/k: k in [1..n]] ): n in [1..45]]; // _G. C. Greubel_, Dec 28 2019
%o A330719 (Sage) [denominator( sum((2^(k-1)-1)/k for k in (1..n)) ) for n in (1..45)] # _G. C. Greubel_, Dec 28 2019
%Y A330719 Cf. A000265, A002805, A064169, A279683, A290348, A330718.
%K A330719 nonn,frac
%O A330719 1,2
%A A330719 _Amiram Eldar_ and _Thomas Ordowski_, Dec 28 2019