A332786 a(n) = numerator(-1/n + Sum_{k=1..n} 2^(k-1)/k).

0, 3, 3, 61, 25, 137, 343, 32663, 2357, 74689, 66671, 5299069, 2416531, 115545821, 106974277, 637525199, 74575583, 1588674349, 4496071973, 3234136824109, 1535024393629, 5843920343363, 5575228585159, 1961561381531581, 93953561866435, 9016382638527647, 2888981280567587, 200248741591132607, 96525489421136333

Offset

1,2

Comments

If p > 3 is a prime, then p^2 | a(p).

Does the above statement follow from Wolstenholme's theorem?

If p is a Wolstenholme prime (A088164), then p^3 | a(p).

However, it should be noted that also 7^3 | a(7).

Conjecture: there are no pseudoprimes m such that m^2 | a(m).

Is 7^2 the only weak pseudoprime (i.e., a composite m such that m | a(m))?

Links

Robert Israel, Table of n, a(n) for n = 1..1367

Formula

a(n) = numerator(-2/n + S(n))/2 for odd n and a(n) = numerator(-2/n + S(n)) for even n, where S(n) = Sum_{k=1..n} 2^k/k, see A108866 / A229726.

a(n) = numerator(Sum_{k=1..n} (2^(k-1)-1)/k + Sum_{k=1..n-1} 1/k), see A330718 / A330719 and A001008 / A002805.

Example

a(5) = numerator(-1/5 + 1/1+2/2+4/3+8/4+16/5) = numerator(128/15 - 1/5) = numerator(25/3) = 25.

Maple

f:= proc(n) local k; numer(-1/n + add(2^(k-1)/k, k=1..n)) end proc:
map(f, [$1..30]); # Robert Israel, Sep 15 2024

Mathematica

n = 30; Numerator[Accumulate @ Table[(2^(k-1))/k, {k, 1, n}] - 1/Range[n]] (* Amiram Eldar, Feb 24 2020 *)

Prog

(PARI) a(n) = numerator(-1/n + sum(k=1, n, 2^(k-1)/k)); \\ Michel Marcus, Feb 24 2020

Crossrefs

Cf. A001008, A088164, A108866, A330718.

Sequence in context: A066807 A165497 A051752 * A102065 A102074 A126227

Adjacent sequences: A332783 A332784 A332785 * A332787 A332788 A332789

Keyword

nonn,frac

Author

Thomas Ordowski, Feb 24 2020

Extensions

More terms from Amiram Eldar, Feb 24 2020

Status

approved