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A330719
a(n) = denominator(Sum_{k=1..n} (2^(k-1) - 1)/k).
6
1, 2, 2, 4, 4, 12, 12, 24, 8, 40, 40, 120, 120, 840, 840, 1680, 1680, 5040, 5040, 5040, 5040, 55440, 55440, 55440, 11088, 144144, 48048, 48048, 48048, 80080, 80080, 160160, 160160, 2722720, 544544, 4900896, 4900896, 93117024, 93117024, 465585120, 465585120, 465585120
OFFSET
1,2
COMMENTS
Conjecture: if p is an odd prime, then p | A330718(p+1) - a(p+1).
Below 10^6 there is only one pseudoprime, namely 25. Are there others?
Primes p such that p^2 | A330718(p+1) - a(p+1) are 3, 5, 45827, ...
LINKS
FORMULA
a(n) = denominator(-(2^n*LerchPhi(2,1,n+1) + Pi*i/2 + HarmonicNumber(n))). - G. C. Greubel, Dec 28 2019
a(n) = denominator(A279683(n)/n!) for n > 0. - Amiram Eldar and Thomas Ordowski, Jan 15 2020
MAPLE
seq(denom(add((2^(k-1)-1)/k, k = 1..n)), n = 1..45); # G. C. Greubel, Dec 28 2019
MATHEMATICA
Denominator@Accumulate@Array[(2^(#-1) -1)/# &, 45]
Table[Denominator[-(2^n*LerchPhi[2, 1, n+1] +Pi*I/2 +HarmonicNumber[n])], {n, 45}] (* G. C. Greubel, Dec 28 2019 *)
PROG
(PARI) a(n) = denominator(sum(k=1, n, (2^(k-1)-1)/k)); \\ Michel Marcus, Dec 28 2019
(Magma) [Denominator( &+[(2^(k-1)-1)/k: k in [1..n]] ): n in [1..45]]; // G. C. Greubel, Dec 28 2019
(Sage) [denominator( sum((2^(k-1)-1)/k for k in (1..n)) ) for n in (1..45)] # G. C. Greubel, Dec 28 2019
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Amiram Eldar and Thomas Ordowski, Dec 28 2019
STATUS
approved