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

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

Offset: 1

Author

Amiram Eldar and Thomas Ordowski, Dec 28 2019

Keywords

• nonn
• frac

Comments

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).

Examples

			Numerators of 0, 1, 3, 13/2, 25/2, 137/6, 245/6, ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Wolstenholme's Theorem

Crossrefs

Cf. A000265, A001008, A007663, A088164, A159353, A225101, A279683, A290347, A330719.

Programs

Magma

[Numerator( &+[(2^k -2)/k: k in [1..n]] ): n in [1..30]]; // G. C. Greubel, Dec 28 2019

Maple

seq(numer(add((2^k -2)/k, k = 1..n)), n = 1..30); # G. C. Greubel, Dec 28 2019

Mathematica

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 *)

PARI

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

Formula

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

a(n+1) = numerator(a(n)/A330719(n) + A225101(n+1)/(2*A159353(n+1))).

a(p) = a(p-1) + A007663(n)*A330719(p-1) for p = prime(n) > 2.

a(n) = numerator(-(2^(n+1)*LerchPhi(2,1,n+1) + Pi*i + 2*HarmonicNumber(n))). - G. C. Greubel, Dec 28 2019

a(n) = numerator(A279683(n)/n!) for n > 0. - Amiram Eldar and Thomas Ordowski, Jan 15 2020

For n > 1, a(n) = A000265(A290347(n)). - Thomas Ordowski, Mar 29 2025