A330719 -id:A330719 - cp's OEIS frontend

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

Amiram Eldar and Thomas Ordowski, Dec 28 2019

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).

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

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

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

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

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

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 *)

a(n) = numerator(sum(k=1, n, (2^k-2)/k)); \\ Michel Marcus, Dec 28 2019

[numerator( sum((2^k -2)/k for k in (1..n)) ) for n in (1..30)] # G. C. Greubel, Dec 28 2019

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

A001901 Successive numerators of Wallis's approximation to Pi/2 (reduced).

Original entry on oeis.org

1, 2, 4, 16, 64, 128, 256, 2048, 16384, 32768, 65536, 262144, 1048576, 2097152, 4194304, 67108864, 1073741824, 2147483648, 4294967296, 17179869184, 68719476736, 137438953472, 274877906944, 2199023255552

Offset: 0

N. J. A. Sloane

If p is prime, then a(p-2) == - A001902(p-2) (mod p). Cf. A064169 (third comment) and my formula here. Such pseudoprimes are 1467, 7831, ... Primes p such that a(p-2) == - A001902(p-2) (mod p^2) are 5, 45827, ... Cf. A355959, see also A330719 (third comment). - Thomas Ordowski, Oct 19 2024

			From _Wolfdieter Lang_, Dec 07 2017: (Start)
The Wallis numerators (N) and denominators (D) with partial products A(n) = A001900(n) and B(n) = A000246(n+1) in unreduced form, and a(n) and b(n) = A001902(n) in reduced form.
n, k:     0  1  2  3  4   5    6     7      8        9       10 ...
N(k):     1  2  2  4  4   6    6     8      8       10       10 ...
D(k):     1  1  3  3  5   5    7     7      9        9        9 ...
A(n):     1  2  4 16 64 384 2304 18432 147456  1474560 14745600 ...
B(n):     1  1  3  9 45 225 1575 11025  99225   893025  9823275 ...
a(n):     1  2  4 16 64 128  256  2048  16384    32768    65536 ...
b(n):     1  1  3  9 45  75  175  1225  11025    19845    43659 ...
n = 5: numerator(1*2*2*4*4*6/(1*1*3*3*5*5)) = numerator(384/225) = numerator(128/75) = 128. (End)

H.-D. Ebbinghaus et al., Numbers, Springer, 1990, p. 146.

Jonathan Sondow, A faster product for Pi and a new integral for ln(Pi/2), arXiv:math/0401406 [math.NT], 2004.
Jonathan Sondow, A faster product for Pi and a new integral for ln(Pi/2), Amer. Math. Monthly 112 (2005), 729-734 and 113 (2006), 670.
Eric Weisstein's World of Mathematics, Pi
Eric Weisstein's World of Mathematics, Pi Continued Fraction
Index to divisibility sequences

Denominators are A001902. Subsequence of A000079.

a[n_?EvenQ] := n!!^2/((n - 1)!!^2*(n + 1)); a[n_?OddQ] := ((n - 1)!!^2*(n + 1))/n!!^2; Table[a[n] // Numerator, {n, 0, 23}] (* Jean-François Alcover, Jun 19 2013 *)

(2*2*4*4*6*6*8*8*...*2n*2n*...)/(1*3*3*5*5*7*7*9*...*(2n-1)*(2n+1)*...) for n >= 1.
From Wolfdieter Lang, Dec 07 2017: (Start)
1/1 * 2/1 * 2/3 * 4/3 * 4/5 * 6/5 * 6/7 * ...; partial products (reduced). Here the numerators with offset 0.
a(n) = numerator(W(n)), for n >= 0, with W(n) = Product_{k=0..n} N(k)/D(k) (reduced), with N(k) = 2*floor((k+1)/2) for k >= 1 and N(0) = 1, and D(k) = 2*floor(k/2) + 1, for k >= 0. (End)
a(n) is the numerator of the continued fraction [1;1,1/2,1/3,...,1/n]. - Thomas Ordowski, Oct 19 2024

A279683 Number of move operations required to sort all permutations of [n] by MTF sort.

Original entry on oeis.org

0, 0, 1, 9, 78, 750, 8220, 102900, 1463280, 23451120, 419942880, 8331634080, 181689298560, 4323472433280, 111534141438720, 3101254066310400, 92468631077222400, 2943141763622860800, 99596858633182310400, 3570677764371119001600, 135190500045467682816000

Offset: 0

Alois P. Heinz, Dec 16 2016

nonn

MTF sort is an (inefficient) sorting algorithm: the first element that is smaller than its predecessor is moved to front repeatedly until the sequence is sorted.

Conjecture: primes p such that p^4 divides a(p) are the Wolstenholme primes A088164. - Amiram Eldar and Thomas Ordowski, Jan 15 2020

			a(0) = a(1) = 0 because 0 or 1 elements are already sorted.
a(2) = 1: [1,2] is sorted and [2,1] needs one move.
a(3) = 9: [1,2,3](0), [1,3,2]->[2,1,3]->[1,2,3](2), [2,1,3]->[1,2,3](1), [2,3,1]->[1,2,3](1), [3,1,2]->[1,3,2]->[2,1,3]->[1,2,3](3), [3,2,1]->[2,3,1]->[1,2,3](2); sum of all moves gives 0+2+1+1+3+2 = 9.

Alois P. Heinz, Table of n, a(n) for n = 0..404
Project Euler, Problem 523: First Sort I
Wikipedia, Sorting algorithm
Index entries for sequences related to sorting

Cf. A000142, A212395, A240797, A280970, A330718, A330719.

a:= proc(n) option remember;
      `if`(n=0, 0, a(n-1)*n + (n-1)! * (2^(n-1)-1))
    end:
seq(a(n), n=0..20);
# second Maple program:
a:= proc(n) option remember; `if`(n<3, [0$2, 1][n+1],
      (4*n-3)*a(n-1)-(n-1)*(5*n-7)*a(n-2)+(2*n-2)*(n-2)^2*a(n-3))
    end:
seq(a(n), n=0..20);

a[0] = 0; a[n_] := a[n] = a[n-1]*n + (n-1)!*(2^(n-1) - 1);
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 30 2018 *)

a(n) = a(n-1)*n + (n-1)! * (2^(n-1)-1) for n>0, a(0) = 0.
a(n) = (4*n-3)*a(n-1)-(n-1)*(5*n-7)*a(n-2)+(2*n-2)*(n-2)^2*a(n-3) for n>2.
a(n) ~ 2^n * (n-1)!. - Vaclav Kotesovec, Dec 25 2016
a(n) = n! * Sum_{k=1..n} (2^(k-1)-1)/k = A000142(n)*A330718(n)/A330719(n), for n > 0. - Amiram Eldar and Thomas Ordowski, Jan 15 2020

A290348 Denominators of the Harary index for the n-halved cube graph.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 3, 1, 5, 5, 15, 15, 105, 105, 105, 105, 315, 315, 315, 315, 3465, 3465, 3465, 693, 9009, 3003, 3003, 3003, 5005, 5005, 5005, 5005, 85085, 17017, 153153, 153153, 2909907, 2909907, 14549535

Offset: 1

Eric W. Weisstein, Jul 28 2017

			First few terms are 0, 1, 6, 26, 100, 1096/3, 3920/3, 13936/3, 16544, 296256/5, ....

Eric Weisstein's World of Mathematics, Halved Cube Graph
Eric Weisstein's World of Mathematics, Hypercube Graph

Cf. A000265, A290347 (numerators), A330719.

Table[-2^(n - 1) HarmonicNumber[n] - 2^(2 n - 1) Re[LerchPhi[2, 1, n + 1]], {n, 20}] // Denominator

a(n) = A000265(A330719(n)). - Thomas Ordowski, Mar 29 2025

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

A331343 a(n) = lcm(1,2,...,n) * Sum_{k=1..n} (2^(k-1) - 1) / k.

Original entry on oeis.org

0, 1, 9, 39, 375, 685, 8575, 30485, 162855, 291627, 5785857, 10514427, 250200951, 461037291, 854622483, 3185234481, 101381371377, 190598779657, 6833215763803, 12935721409039, 24559552771039, 46750514134519, 2051664357879617, 3923102768811707, 37581323659852375

Offset: 1

Amiram Eldar and Thomas Ordowski, Jan 14 2020

nonn

By Wolstenholme's theorem, if p > 3 is a prime, then p^3 | a(p).
Conjecture: for n > 3, if n^3 | a(n), then n is prime. If so, there are no such pseudoprimes.
Problem: are there weak pseudoprimes m such that m^2 | a(m)? None up to 5*10^4.
Composite numbers m such that m | a(m) are 9, 25, 49, 99, 121, 125, 169, 221, 289, 343, 357, 361, 399, 529, 665, 841, 961, 1331, 1369, 1443, 1681, 1849, 2183, ...  Cf. A082180.
Prime numbers p such that p^4 | a(p) are probably only the Wolstenholme primes A088164.

Wikipedia, Wolstenholme's theorem.

Cf. A003418, A025529, A082180, A088164, A330718, A330719.

[Lcm([1..n])*&+[(2^(k-1)-1)/k:k in [1..n]]:n in [1..25]]; // Marius A. Burtea, Jan 14 2020

a[n_] := LCM @@ Range[n] * Sum[(2^(k-1) - 1) / k, {k, 1, n}]; Array[a, 25]

a(n) = lcm([1..n])*sum(k=1, n, (2^(k-1) - 1) / k); \\ Michel Marcus, Jan 14 2020

a(n) = A003418(n) * A330718(n) / A330719(n).

cp's OEIS Frontend

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

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A001901 Successive numerators of Wallis's approximation to Pi/2 (reduced).

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A279683 Number of move operations required to sort all permutations of [n] by MTF sort.

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A290348 Denominators of the Harary index for the n-halved cube graph.

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

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A331343 a(n) = lcm(1,2,...,n) * Sum_{k=1..n} (2^(k-1) - 1) / k.

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