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

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

Author

Amiram Eldar and Thomas Ordowski, Dec 28 2019

Keywords

• nonn
• frac

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, ...

Examples

			Denominators of 0, 1/2, 3/2, 13/4, 25/4, 137/12, 245/12, ...
		

Links

• Metin Sariyar, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A000265, A002805, A064169, A279683, A290348, A330718.

Programs

Magma

[Denominator( &+[(2^(k-1)-1)/k: k in [1..n]] ): n in [1..45]]; // G. C. Greubel, Dec 28 2019

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 *)

PARI

a(n) = denominator(sum(k=1, n, (2^(k-1)-1)/k)); \\ Michel Marcus, 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

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

A000265(a(n)) = A290348(n). - Thomas Ordowski, Mar 29 2025