A108866 -id:A108866 - cp's OEIS frontend

A229726 Denominator of Sum_{k=1..2n+1} 2^k/k.

1, 3, 15, 105, 315, 3465, 45045, 9009, 153153, 2909907, 14549535, 334639305, 1673196525, 5019589575, 145568097675, 4512611027925, 4512611027925, 4512611027925, 166966608033225, 166966608033225, 6845630929362225, 294362129962575675, 294362129962575675, 13835020108241056725, 96845140757687397075, 96845140757687397075, 5132792460157432044975

Offset: 0

N. J. A. Sloane, Sep 28 2013

Numerator is A108866.

			2, 20/3, 256/15, 4832/105, 42496/315, 1467392/3465, 62836736/45045, 42790912/9009, 2535587840/153153, 170851041280/2909907, 3070050172928/14549535, 256460544016384/334639305, 4697678227177472/1673196525, 52001663422038016/5019589575, ...

Hugo Pfoertner, Table of n, a(n) for n = 0..500

Cf. A229727, A108866.

for(n=0,26,print1(denominator(sum(k=1,2*n+1,2^k/k)),", ")) \\ Hugo Pfoertner, Mar 07 2020

A087910 Exponent of the greatest power of 2 dividing the numerator of 2^1/1 + 2^2/2 + 2^3/3 + ... + 2^n/n.

Original entry on oeis.org

1, 2, 2, 5, 8, 5, 5, 13, 9, 10, 10, 12, 12, 12, 12, 22, 17, 18, 18, 21, 22, 21, 21, 27, 25, 26, 26, 27, 27, 27, 27, 40, 33, 34, 34, 37, 39, 37, 37, 48, 41, 42, 42, 44, 44, 44, 44, 54, 49, 50, 50, 53, 54, 53, 53, 58, 57, 59, 62, 58, 58, 58

Offset: 1

Robin Chapman, Oct 17 2003

Problem 9 of the 2002 Sydney University Mathematical Society Problems competition asked for a proof that a(n) tends to infinity with n. While this is immediate from the theory of the 2-adic logarithm, elementary proofs are available.

a(n) tends to infinity with n implies that log(-1) = 0 in the 2-adic field, by setting x = 2 in -log(1-x) = Sum_{k>=1} x^k/k. - Jianing Song, Aug 05 2019

			a(5) = 8 as 2^1/1 + 2^2/2 + 2^3/3 + 2^4/4 + 2^5/5 = 256/15 whose numerator is divisible by 2^8 but not by 2^9.

A. M. Robert, A Course in p-adic Analysis, Springer, 2000; see p. 278.

Robert Israel, Table of n, a(n) for n = 1..5000
Sydney University Mathematical Society Problems Competitions, see Problem 9 of the 2002 competition.

Cf. A108866, A007814, A119387, A000523.

S:= 0:
for n from 1 to 100 do
  S:= S + 2^n/n;
  a[n]:= padic:-ordp(numer(S),2);
od:
seq(a[n],n=1..100); # Robert Israel, Jun 09 2015

s[n_] := -2^(n + 1) LerchPhi[2, 1, n + 1] - I Pi;
a[n_] := IntegerExponent[Numerator[Simplify[s[n]]], 2];
Array[a, 62] (* Peter Luschny, Feb 22 2020 *)

a(n) = valuation(sum(k=1,n,2^k/k), 2) \\ Jianing Song, Feb 22 2020

a(n) = A007814(A108866(n)). - Michel Marcus, Feb 22 2020

Sum_{k=1..n} 2^k/k = (2^n/n)*Sum_{k=0..n-1} 1/binomial(n-1,k), so a(n) >= n - v(n,2) - max_{k=0..n-1} v(binomial(n-1,k),2) = n - A007814(n) - A119387(n) = n - floor(log_2(n)), where v(n,2) is the 2-adic valuation of n. It seems that the equality holds if and only if n = 2^m - 1 for some m. - Jianing Song, Feb 22 2020

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

Original entry on oeis.org

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

Thomas Ordowski, Feb 24 2020

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))?

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

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

Cf. A001008, A088164, A108866, A330718.

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

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

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

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.

More terms from Amiram Eldar, Feb 24 2020

A229727 Numerator of Sum_{k=1..2n+1} 2^k/k.

Original entry on oeis.org

2, 20, 256, 4832, 42496, 1467392, 62836736, 42790912, 2535587840, 170851041280, 3070050172928, 256460544016384, 4697678227177472, 52001663422038016, 5598478396465086464, 647670579588837146624, 2427980359983876276224, 9138219784535816536064, 1277040945143933271277568

Offset: 0

N. J. A. Sloane, Sep 28 2013

			2, 20/3, 256/15, 4832/105, 42496/315, 1467392/3465, 62836736/45045, 42790912/9009, 2535587840/153153, 170851041280/2909907, 3070050172928/14549535, 256460544016384/334639305, 4697678227177472/1673196525, 52001663422038016/5019589575, ...

Cf. A229726, A108866.

Take[Numerator[Accumulate[Table[2^k/k,{k,51}]]],{1,-1,2}] (* Harvey P. Dale, Jun 28 2021 *)

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A229726 Denominator of Sum_{k=1..2n+1} 2^k/k.

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A087910 Exponent of the greatest power of 2 dividing the numerator of 2^1/1 + 2^2/2 + 2^3/3 + ... + 2^n/n.

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

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A229727 Numerator of Sum_{k=1..2n+1} 2^k/k.

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