Anthony C Robin - cp's OEIS frontend

A224231 Egyptian fraction expansion of sqrt(3).

1, 2, 5, 32, 1249, 5986000, 438522193400489, 3126430743599145840898147625516, 10008815260914521335142941393259537613217919681721512170785592

Offset: 0

N. J. A. Sloane, Apr 11 2013, following a suggestion from Anthony C Robin

nonn

			sqrt(3) = 1 + 1/2 + 1/5 + 1/32 + 1/1249 + 1/5986000 + ...

A118325 is the main entry for this sequence.

A224230 Egyptian fraction expansion of Pi.

Original entry on oeis.org

3, 8, 61, 5020, 128541455, 162924332716605980, 28783052231699298507846309644849796, 871295615653899563300996782209332544845605756266650946342214549769447

Offset: 0

N. J. A. Sloane, Apr 11 2013, following a suggestion from Anthony C Robin

nonn

A variant of A001466, which is the main entry.

			Pi = 3 + 1/8 + 1/61 + 1/5020 + ...

Cf. A001466, A182257.

A217948 List of numbers 2n for which the riffle permutation permutes all except the first and last of the 2n cards.

Original entry on oeis.org

4, 6, 12, 14, 20, 30, 38, 54, 60, 62, 68, 84, 102, 108, 132, 140, 150, 164, 174, 180, 182, 198, 212, 228, 270, 294, 318, 348, 350, 374, 380, 390, 420, 422, 444, 462, 468, 492, 510, 524, 542, 548, 558, 564, 588, 614, 620, 654, 660, 662, 678, 702, 710, 758, 774, 788, 798

Offset: 1

N. J. A. Sloane, Nov 07 2012, based on an email message from Anthony C Robin

nonn

With 2n cards, a riffle shuffle can be described as a permutation, where r becomes 2r-1 when r <= n and r becomes 2r-2n when r > n. The first and last cards are always left unaltered. Sequence A002326 describes the lengths of the longest orbits in the permutation. E.g. when 2n=10, the permutation can be described as (2,3,5,9,8,6)(4,7). The present sequence gives the values of 2n for which there is just one orbit on the 2n-2 cards, for example the permutation when 2n=12 is (2,3,5,9,6,11,10,8,4,7) containing all the 10 numbers other than 1 & 12.

Tiago Januario (email, Jan 12 2015; see also reference) conjectures that these terms are always one more than a prime. - N. J. A. Sloane, Mar 02 2015

Tiago Januario and Sebastian Urrutia, An Analytical Study in Connectivity of Neighborhoods for Single Round Robin Tournaments, 14th INFORMS Computing Society Conference, Richmond, Virginia, January 11-13, 2015, pp. 188-199; http://dx.doi.org/10.1287/ics.2015.0014
Tiago Januario, S Urrutia, D de Werra, Sports scheduling search space connectivity: A riffle shuffle driven approach, Discrete Applied Mathematics, Volume 211, 1 October 2016, Pages 113-120; http://dx.doi.org/10.1016/j.dam.2016.04.018

Olivier Gérard and Vincenzo Librandi, Table of n, a(n) for n = 1..6000 (first 386 terms from Olivier Gérard).
Sebastián Urrutia, Dominique de Werra, and Tiago Januario, Recoloring subgraphs of K_(2n) for Sports Scheduling, Theoretical Computer Science (2021) Vol. 877, 36-45.

Equals twice A051733.

Cf. A002326, A051732.

(* v8 *)  2*Select[Range[2,1000],Function[n,Sort[First[First[ PermutationCycles@Join[Table[2r-1,{r,1,n}],Table[2r-2n,{r,n+1,2n}]]]]]== Range[2,2n-1]]] (* Olivier Gérard, Nov 08 2012 *)

From Joerg Arndt, Dec 15 2012: (Start)
Apparently a(n) = A179194(n) - 1.
a(n) = 2 * A051733(n). (End)

A178145 6n-1,6n+1, 6n+5, 6n+7 are all primes. That is they are adjacent pairs of twin primes.

Original entry on oeis.org

1, 2, 17, 32, 137, 247, 312, 347, 542, 577, 942, 1572, 2167, 2607, 2622, 2677, 3007, 3152, 3237, 3502, 3712, 4217, 5287, 5807, 7297, 8557, 9222, 10497, 11202, 11582, 12037, 12877, 13282, 13507, 13787, 14802, 16307, 16522, 16852, 18307, 19422

Offset: 1

Anthony C Robin, May 21 2010

nonn

The sequence A002822, gives the values of n to generate the twin primes, 6n+/-1. It has the values 1, 2, 3, 5, 7, 10, 12, 17, 18, 23, . This new sequence lists the lower of the values of this which are adjacent, namely 1(&2), 2 (&3), 17 (&18)..

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A002822

(First[#]+1)/6&/@Select[Partition[Prime[Range[12000]],4,1], Differences[#] == {2,4,2}&] (* Harvey P. Dale, Jan 23 2012 *)

Extended by D. S. McNeil, May 23 2010

A140650 Number of different ways of coloring an n X n grid of squares using two colors so that the resulting grid has just one line of symmetry.

Original entry on oeis.org

0, 3, 48, 600, 32256, 1177344, 268369920, 36506664960, 35184338534400, 18577347909255168, 73786976226118729728, 153476910691030086451200, 2475880078570197599844827136, 20440865928680162788862343512064, 1329227995784915854457062986570792960

Offset: 1

Anthony C Robin, Jul 09 2008

nonn

Cf. A054257, A054407.

s=[0]; for(n=1, 10, s=concat(s, [2^(2*n^2-1)*(2^n+1)-2^(n^2-1)*(2^n+1), 2^(2*n^2+3*n+1)-2^(n^2+2*n+1)])); s \\ Colin Barker, Mar 28 2014

a(2n) = 2^(2n^2-1)*(2^n+1)-2^(n^2-1)*(2^n+1).

a(2n+1) = 2^(2n^2+3n+1)-2^(n^2+2n+1). (corrected by Colin Barker, Mar 28 2014)

More terms from Colin Barker, Mar 28 2014

A140789 We consider how many ways there are of coloring a square grid, n X n, using just two colors, black & white say. If the resulting grid has TWO lines of symmetry, then the number of different grids is given by this sequence. None of these counted are the images of any of the others under a rotation. If one wishes to count these as different, then each of these numbers can be multiplied by 2.

Original entry on oeis.org

0, 1, 8, 32, 448, 2240, 64512, 556032, 33521664, 553615360, 68717379584, 2233380896768, 562949684985856, 36310271727239168, 18446744004990074880, 2370406613402957905920, 2417851639194073977323520, 620178945462269582252179456, 1267650600228193372699684241408

Offset: 1

Anthony C Robin, Jul 14 2008

nonn

Cf. A140650.

s=[0]; for(m=1, 15, s=concat(s, [2^(m^2-1)*(2^m+1)-2^(m*(m+1)/2), 2^(m^2+2*m+1)-2^((m^2+3*m+2)/2)])); s \\ Colin Barker, Mar 28 2014

a(2m) = 2^(m^2 - 1)*(2^m + 1) - 2^(m*(m + 1)/2).

a(2m+1)= 2^(m^2 + 2*m + 1) - 2^((m^2 + 3*m + 2)/2).

More terms from Colin Barker, Mar 28 2014

A140795 We consider how many ways there are of coloring a square grid, n X n, using just two colors, black & white say. If the resulting grid has rotational symmetry of order two only, then the number of different grids is given by this sequence. None of these counted are the images of any of the others under a reflection or a rotation of 90 degrees. If one wishes to count these as different, then each of these numbers can be multiplied by 4.

Original entry on oeis.org

0, 0, 2, 44, 1792, 64288, 8354304, 1073447424, 549738528768, 281474691514368, 576460717407862784, 1180591619583540985856, 9671406556633359531900928, 79228162514246041720191975424, 2596148429267404554864448650608640, 85070591730234614676028659138035712000

Offset: 1

Anthony C Robin, Jul 15 2008

nonn

Cf. A054247, A054407.

s=[0]; for(m=1, 15, s=concat(s, [(2^(2*m^2)-2^m^2*(2^m+2)+2^((m^2+m+2)/2))/4, 2^(2*m^2+2*m-1)-2^(m^2-1)*(2^(2*m+1)+2^m)+2^(m*(m+3)/2)])); s \\ Colin Barker, Mar 28 2014

a(2m+1) = 2^(2*m^2 + 2*m - 1) - 2^(m^2 - 1)*(2^(2*m + 1) + 2^m) + 2^(m*(m + 3)/2).

a(2m) = (2^(2*m^2) - 2^m^2*(2^m + 2) + 2^((m^2 + m + 2)/2))/4.

More terms from Colin Barker, Mar 28 2014

A153194 Numbers such that the numerator of floor(sqrt(n))/n, when reduced to its lowest terms, is equal to 3.

Original entry on oeis.org

10, 11, 13, 14, 38, 40, 44, 46, 84, 87, 93, 96, 148, 152, 160, 164, 230, 235, 245, 250, 330, 336, 348, 354, 448, 455, 469, 476, 584, 592, 608, 616, 738, 747, 765, 774, 910, 920, 940, 950, 1100, 1111, 1133, 1144, 1308, 1320, 1344, 1356

Offset: 1

Anthony C Robin, Dec 20 2008

Previous name was: "For each number of the sequence, n, consider the fraction of squares from 1 to n inclusive. For numbers in this sequence, that fraction, when reduced to its lowest terms will always have 3 in the numerator.".

To obtain similar fractions as above with a numerator of 1, for example 1/5 are square, there are three possible numbers, namely 15, 20, 25. In general it is fairly easy to show that for 1/k of the numbers up to n (1 to n inclusive) to be square, n takes one of the three values, k(k-2), k(k-1), k^2. This sequence looks at obtaining fractions of the form 3/k. Another sequence (A153192) looks at the 2/k case.

			For 38, there are 6 squares below it and 6/38=3/19.
For 164 there are 12 squares below it and 12/164=3/41.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1).

Cf. A153192.

CoefficientList[Series[(10 + x + 2 x^2 + x^3 + 4 x^4)/((1 - x) (1 - x^4)^2), {x, 0, 60}], x] (* Vincenzo Librandi, Mar 29 2014 *)

isok(n) = numerator(sqrtint(n)/n) == 3 \\ Michel Marcus, Aug 05 2013

Vec(-x*(4*x^4+x^3+2*x^2+x+10)/((x-1)^3*(x+1)^2*(x^2+1)^2) + O(x^100)) \\ Colin Barker, Mar 28 2014

G.f.: -x*(4*x^4+x^3+2*x^2+x+10) / ((x-1)^3*(x+1)^2*(x^2+1)^2). - Colin Barker, Mar 28 2014
G.f.: x * (10 + x + 2*x^2 + x^3 + 4*x^4) / ((1 - x) * (1 - x^4)^2). - Michael Somos, Mar 28 2014
a(n+4) = 2*a(n) - a(n-4) + 18 if n>0. - Michael Somos, Mar 28 2014

Edited and more terms added by Michel Marcus, Aug 05 2013

A153192 Numbers such that the numerator of floor(sqrt(n))/n, when reduced to its lowest terms, is equal to 2.

Original entry on oeis.org

5, 7, 18, 22, 39, 45, 68, 76, 105, 115, 150, 162, 203, 217, 264, 280, 333, 351, 410, 430, 495, 517, 588, 612, 689, 715, 798, 826, 915, 945, 1040, 1072, 1173, 1207, 1314, 1350, 1463, 1501, 1620, 1660, 1785, 1827, 1958

Offset: 1

Anthony C Robin, Dec 20 2008

Previous name was: "For each term in this sequence, n say, consider the fraction of square numbers between 1 & n, inclusive, when reduced to its lowest terms. This fraction will always have a numerator of 2 for numbers in this sequence".
To obtain similar fractions as above with a numerator of 1, for example 1/5 are square, there are three possible numbers, namely 15, 20, 25. In general it is fairly easy to show that for 1/k of the numbers up to n (1 to n inclusive) to be square, n takes one of the three values, k(k-2), k(k-1), k^2. This sequence looks at obtaining fractions of the form 2/k. Another sequence (A153194) looks at the 3/k case.
Alternately, numbers of the form 4n^2+n or 4n^2+3n. - Charles R Greathouse IV, Aug 05 2013

			22 has 4 square numbers below it and 4/22=2/11.
76 has 8 square numbers below it and 8/76=2/19.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A153194.

LinearRecurrence[{1,2,-2,-1,1},{5,7,18,22,39},50] (* Harvey P. Dale, Sep 23 2022 *)

isok(n) = numerator(sqrtint(n)/n) == 2 \\ Michel Marcus, Aug 05 2013

for(n=1,9,print1(4*n^2+n", "4*n^2+3*n", ")) \\ Charles R Greathouse IV, Aug 05 2013

From Colin Barker, Mar 28 2014: (Start)
a(n) = (2*n+3)*(2*n-(-1)^n+1)/4.
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5).
G.f.: -x*(x^2+2*x+5) / ((x-1)^3*(x+1)^2). (End).

Definition simplified and more terms added by Michel Marcus, Aug 05 2013

A133225 Largest prime <= 2^((n+1)/2).

Original entry on oeis.org

2, 2, 3, 5, 7, 11, 13, 19, 31, 43, 61, 89, 127, 181, 251, 359, 509, 719, 1021, 1447, 2039, 2887, 4093, 5791, 8191, 11579, 16381, 23167, 32749, 46337, 65521, 92681, 131071, 185363, 262139, 370723, 524287, 741431, 1048573, 1482907, 2097143, 2965819

Offset: 1

Anthony C Robin, Jan 03 2008

nonn

If one is trying to decide whether an n+1 digit binary number is prime, this is the largest prime for which one needs to test divisibility. For example a six digit number like 110101 must be below 64, so only primes up to 7 are needed to test divisibility. Compare with sequence A132153.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A132153.

seq(prevprime(floor(2^((n+1)*1/2))+1),n=1..40); # Emeric Deutsch
A017910 := proc(n) floor(2^(n/2)) ; end: A007917 := proc(n) prevprime(n+1) ; end: A133225 := proc(n) A007917(A017910(n+1)) ; end: seq(A133225(n),n=1..60) ; # R. J. Mathar

PrevPrim[n_] := Block[{k = n}, While[ !PrimeQ@k, k-- ]; k]; f[n_] := PrevPrim@ Floor@ Sqrt[2^(n + 1)]; Array[f, 42] (* Robert G. Wilson v *)
Table[Prime[PrimePi[2^((n + 1)/2)]], {n, 1, 50}] (* Stefan Steinerberger *)
lp[n_]:=Module[{c=2^((n+1)/2)},If[PrimeQ[c],c,NextPrime[c,-1]]]; Array[lp,50] (* Harvey P. Dale, Aug 25 2013 *)

a(n) = A007917[A017910(n+1)]. - R. J. Mathar

More terms from Stefan Steinerberger, R. J. Mathar, Robert G. Wilson v and Emeric Deutsch, Jan 06 2008

cp's OEIS Frontend

User: Anthony C Robin

A224231 Egyptian fraction expansion of sqrt(3).

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A224230 Egyptian fraction expansion of Pi.

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A217948 List of numbers 2n for which the riffle permutation permutes all except the first and last of the 2n cards.

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A178145 6n-1,6n+1, 6n+5, 6n+7 are all primes. That is they are adjacent pairs of twin primes.

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A140650 Number of different ways of coloring an n X n grid of squares using two colors so that the resulting grid has just one line of symmetry.

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PARI

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A153194 Numbers such that the numerator of floor(sqrt(n))/n, when reduced to its lowest terms, is equal to 3.

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A153192 Numbers such that the numerator of floor(sqrt(n))/n, when reduced to its lowest terms, is equal to 2.

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