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A304231 Numbers of the form m*k with m <= k < 2m.

1, 4, 6, 9, 12, 15, 16, 20, 24, 25, 28, 30, 35, 36, 40, 42, 45, 48, 49, 54, 56, 60, 63, 64, 66, 70, 72, 77, 80, 81, 84, 88, 90, 91, 96, 99, 100, 104, 108, 110, 112, 117, 120, 121, 126, 130, 132, 135, 140, 143, 144, 150, 153, 154, 156, 160, 165, 168, 169, 170

Offset: 1

Keenan Pepper, May 08 2018

From Hartmut F. W. Hoft, Nov 04 2022: (Start)
Three equivalent properties that describe this sequence:
(1) Numbers j satisfying { (m, k) : j = m*k and m <= k < 2*m } != { } -- definition of the sequence.
(2) Numbers j satisfying { d : d | j and sqrt(j/2) < d < sqrt(2*j) } != { } -- stricter than middle divisors.
(3) Numbers j satisfying { d : d | j and d, j/d <= r(j) } != { } -- r(j) = floor((sqrt(8*j+1)-1)/2).
Computations using property (2) are significantly slower than those using properties (1) or (3). (End)

			From _Hartmut F. W. Hoft_, Nov 04 2022: (Start)
72 = 2*6^2 is in this sequence since it has divisors 8 and 9 between 6 and 12.
50 = 2*5^2 is not in this sequence since it has no divisors between 5 and 10.
180 = 2^2 * 3^2 * 5 has the 11 divisors 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18 less than or equal to 18 = r(180), but only the 7 divisors 20, 30, 36, 45, 60, 90, 180 greater than 18. Since sqrt(90) < 10 < 12 < 15 < 18 = r(180) < sqrt(360) and 10 < 18 < 20 and 12 < 15 < 24, all three properties stated above are demonstrated. (End)

Hartmut F. W. Hoft, Proof of the equivalences.

Slightly more strict than A071562 -- only some terms of the form 2*j^2 are omitted.

(* implementation of property (1) *)
a304231[n_] := Module[{list={}, i, j}, For[i=1, i<=Sqrt[n], i++, j=i; While[i j<=n&&j<2i, AppendTo[list, i j]; j++]]; Union[list]]
a304231[170] (* Hartmut F. W. Hoft, Nov 04 2022 *)

isok(n) = fordiv(n, d, if ((d >= n/d) && (d < 2*n/d), return (1))); \\ Michel Marcus, May 25 2018

sorted(sum([[i*j for j in range(i,2*i)] for i in range(100)], []))

A227405 Order of the symmetry group of the densest possible packing of N circles in a larger circle. (If there are different patterns with the same density pick the largest value of the symmetry group order.)

Original entry on oeis.org

4, 6, 8, 10, 12, 12, 14, 16, 2, 2, 6, 2

Offset: 2

Keenan Pepper, Sep 21 2013

a(14) is unknown but conjectured to be 2.

The conjectured sequence is 4, 6, 8, 10, 12, 12, 14, 16, 2, 2, 6, 2, 2, 10, 2, 2, 12, 12, 2...

Wikipedia, Circle packing in a circle

A206788 Denominators of semiconvergents to log_2(3), which equals log(3)/log(2).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 12, 17, 29, 41, 53, 94, 147, 200, 253, 306, 359, 665, 971, 1636, 2301, 2966, 3631, 4296, 4961, 5626, 6291, 6956, 7621, 8286, 8951, 9616, 10281, 10946, 11611, 12276, 12941, 13606, 14271, 14936, 15601, 16266, 31867, 47468, 79335, 111202, 190537

Offset: 1

Keenan Pepper, Feb 12 2012

These are also equal divisions of the octave that have good approximations of the 3rd harmonic (and hence other Pythagorean intervals such as 3/2, 4/3, 9/8 etc.).

The semiconvergents are important musically as the cardinalities of MOS scales of Pythagorean tuning.

Wikipedia, Definition of continued fraction semiconvergents

Cf. A028507 (continued fraction expansion), A005664 (denominators of convergents).

A190482 Convex, obtuse, hexagonal lattice numbers.

Original entry on oeis.org

7, 10, 12, 13, 14, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61

Offset: 1

Keenan Pepper, May 24 2011

These are the integers N for which it is possible to arrange N points in a subset of a hexagonal lattice such that:
1. The points are not all collinear.
2. The convex hull of the points consists entirely of line segments between nearest-neighbor lattice points in the set. (In other words, the natural polygon you get by following nearest-neighbor connections around the outside of the set is a convex polygon.)
3. The convex hull contains only obtuse angles, which will necessarily be 120 degrees.
These can also be thought of as "convenient bundle of cylinders" numbers, because if you are carrying a bundle of N parallel, identical cylinders, only if N is in this list can they be arranged in such a way that there are no "gaps", and also no single cylinders sticking out of the pattern creating "bulges".
The greatest integer that is absent from this list is 17. (This can be proved by exhaustion - it's possible to find infinite sequences of patterns that eventually cover every possible remainder of N mod 6, and 17 is the last N that cannot be achieved.)

			For N = 2, 3, 4, 5, or 6, it is impossible to arrange N cylinders in a neat bundle with no bulges or dents, so they are not in the list. But for N = 7, six cylinders can surround a central one, so 7 is in the list.
It's unclear whether 1 should be included, but the strict definition given above excludes N = 1.

a(n)=if(n>6,n+11,[7,10,12,13,14,16][n]) \\ Charles R Greathouse IV, Aug 26 2011

A180604 Number of "pieces" of the n-th Brillouin zone of a 2D hexagonal lattice.

Original entry on oeis.org

1, 6, 6, 6, 12, 18, 30, 30, 18, 18, 36, 42, 54, 54, 42, 66

Offset: 1

Keenan Pepper, Sep 11 2010

Each Brillouin zone has the same area, and they are all equivalent mod the periodicity of the lattice, but the higher-order ones consist of a set of polygons. This is the number of polygons in that set.

			The 2nd Brillouin zone consists of 6 equilateral triangles, so a(2) = 6. The 5th Brillouin zone consists of 12 30-60-90 triangles, so a(5) = 12.

A180603 Number of "pieces" of the n-th Brillouin zone of a 2D square lattice.

Original entry on oeis.org

1, 4, 8, 12, 20, 20, 12, 12, 20, 28, 44, 48, 48

Offset: 1

Keenan Pepper, Sep 11 2010

Each Brillouin zone has the same area, and they are all equivalent mod the periodicity of the lattice, but the higher-order ones consist of a set of polygons. This is the number of polygons in that set.

			The 2nd Brillouin zone consists of 4 right isosceles triangles, so a(2) = 4. The 4th Brillouin zone consists of 8 triangles and 4 kites, so a(4) = 12.

Jaroslaw W. Klos, 2D Brillouin Zones, Wolfram Demonstrations Project, 2013.

Cf. A180604, A199765, A199766.

A163252 a(0) = 0, and a(n) is the least positive integer not occurring earlier in the sequence such that a(n-1) and a(n) differ in only one bit when written in binary.

Original entry on oeis.org

0, 1, 3, 2, 6, 4, 5, 7, 15, 11, 9, 8, 10, 14, 12, 13, 29, 21, 17, 16, 18, 19, 23, 22, 20, 28, 24, 25, 27, 26, 30, 31, 63, 47, 39, 35, 33, 32, 34, 38, 36, 37, 45, 41, 40, 42, 43, 59, 51, 49, 48, 50, 54, 52, 53, 55, 119, 87, 71, 67, 65, 64, 66, 70, 68, 69, 77, 73, 72, 74, 75, 79, 78

Offset: 0

Keenan Pepper, Jul 23 2009

Step from a(488) = 237 = 11101101_2 to a(489) = 749 = 1011101101_2 is the first case when one term is two binary digits longer than the previous. Considering the leading zeros, though, they still differ in only one bit. - Ivan Neretin, Jun 25 2015

Paul Tek, Table of n, a(n) for n = 0..10000

Cf. A003188, A101080, A360307 (inverse).

N:= 10: # to get all terms before the first where a(n) >= 2^N
B:= Array(0..2^N-1):
B[0]:= 1:
a[0]:= 0:
L:= Vector([0$N]):
for n from 1 do
  cands:= select(t -> B[t[1]]=0, [seq(`if`(L[i]=0,[a[n-1]+2^(i-1),i],[a[n-1]-2^(i-1),i]),i=1..N)]);
  if nops(cands)=0 then break fi;
  j:= min[index](map(t->t[1],cands));
  a[n]:= cands[j][1];
  i:= cands[j][2];
  B[a[n]]:= 1;
  L[i]:= 1 - L[i];
od:
seq(a[i],i=0..n-1); # Robert Israel, Jun 25 2015

Nest[Append[#, Min[Complement[BitXor[#[[-1]], 2^Range[0, Floor[Log2[#[[-1]]]] + 2]], #]]] &, {0, 1}, 71] (* Ivan Neretin, Jun 25 2015 *)

From Alois P. Heinz, Feb 02 2023: (Start)
A101080(a(n),a(n+1)) = 1.
a(A360307(n)) = n = A360307(a(n)). (End)

A133587 Conjectured order of the symmetry group of the (numerically computed) least-perimeter cluster of n nonoverlapping circles.

Original entry on oeis.org

4, 6, 4, 2, 10, 12, 14, 2, 4, 2, 6, 2, 4, 1, 2, 2, 2, 12, 2, 2, 1

Offset: 2

Keenan Pepper, Dec 27 2007

nonn

This can be thought of as the order of the symmetry group of the minimum-energy configuration of n two-dimensional bubbles in a plane. a(1) is infinite, because one bubble forms a circle, which has a continuous symmetry group containing rotations of arbitrary angles. So far, the actual symmetry groups are all dihedral, except for a(15) and a(22), which are trivial (their configurations have no symmetries).

			a(3) = 6 because three planar bubbles arrange themselves in an equilateral-triangle-type configuration with symmetry group D_3, of order 6.

Cox, S. J., F. Graner, M. F. Vaz, C. Monnereau-Pittet and N. Pittet, 2003, Minimal perimeter for N identical bubbles in two dimensions: calculations and simulations, Philos. Mag. 83, 1393-1406.
F. Morgan, Soap bubble clusters, Rev. Mod. Phys. Vol. 79 (2007), pp. 821-827.

R. L. Graham and N. J. A. Sloane, Penny-Packing and Two-Dimensional Codes, Discrete and Comput. Geom. 5 (1990), 1-11.

Cf. A133491.

A133491 Order of the symmetry group of the (in some cases conjectural) minimal-energy configuration of n identical charged particles confined to the surface of a sphere.

Original entry on oeis.org

12, 24, 12, 48, 20, 16, 12, 16, 4, 120, 4

Offset: 3

Keenan Pepper, Nov 30 2007

a(0), a(1) and a(2) are all infinite, because their symmetry groups are continuous (they contain rotations with arbitrary angles). Actual symmetry groups: 3 D_{3h}, 4 T_{d}, 5 D_{3h}, 6 O_{d}, 7 D_{5h}, 8 D_{4d}, 9 D_{3h}, 10 D_{4h}, 11 D_{1h}, 12 I_{d}, 13 D_{1h}.

			a(3)=12 because the minimal-energy configuration of 3 charged particles on a sphere is an equilateral triangle on the equator, which has symmetry group D_3h of order 12.

K. S. Brown, Min-Energy Configurations of Electrons On A Sphere, MathPages.
R. H. Hardin, N. J. A. Sloane and W. D. Smith, Minimal Energy Configurations of Points on a Sphere
Wikipedia, Thomson Problem.

Cf. A033177, A242617.

A116474 Equal divisions of the octave with progressively increasing consistency levels.

Original entry on oeis.org

1, 3, 4, 5, 22, 26, 29, 58, 80, 94, 282, 311, 17461

Offset: 3

Keenan Pepper, Mar 17 2006

An equal temperament is consistent at level N (odd integer) if all the intervals in the N-limit tonality diamond (set of ratios with odd factors of numerator and denominator not exceeding N) are approximated consistently, i.e. the composition of the approximations is the closest approximation of the composition.
These EDOs are not necessarily any good for musical purposes. Even though 4-EDO is consistent through the 7 limit, no one would seriously consider using it for 7-limit music because the approximations are so bad.
While for the smallest values these EDOs are not directly usable, their consistency is even so a valuable feature. For example, 4-EDO is consistent through the 7 limit, but is not usable directly for 7-limit music. However, indirectly, by means of subsequently adjusting the harmony, it can be and has been useful as a compositional tool for composing music in the 7-limit. The same comment applies to 3 in the 5-limit and 5 in the 9-limit. Any of the values above 5 are usable directly as equal temperament approximations to the corresponding limit. - Gene Ward Smith, Mar 29 2006

			3-EDO is consistent through the 5 limit because 6/5, 5/4 and 4/3 map to 1 step and 3/2, 8/5 and 5/3 map to 2 steps and all the compositions work out, for example 6/5 * 5/4 = 3/2 and 1 step + 1 step = 2 steps. It is not consistent through the 7 limit because 8/7 and 7/6 both map to 1 step, but 8/7 * 7/6 = 4/3 also maps to 1 step.

Cf. A116475, A117577, A117578.

More terms from Gene Ward Smith, Mar 29 2006

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User: Keenan Pepper

A304231 Numbers of the form m*k with m <= k < 2m.

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A227405 Order of the symmetry group of the densest possible packing of N circles in a larger circle. (If there are different patterns with the same density pick the largest value of the symmetry group order.)

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A206788 Denominators of semiconvergents to log_2(3), which equals log(3)/log(2).

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A190482 Convex, obtuse, hexagonal lattice numbers.

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A180604 Number of "pieces" of the n-th Brillouin zone of a 2D hexagonal lattice.

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A180603 Number of "pieces" of the n-th Brillouin zone of a 2D square lattice.

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A163252 a(0) = 0, and a(n) is the least positive integer not occurring earlier in the sequence such that a(n-1) and a(n) differ in only one bit when written in binary.

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A133587 Conjectured order of the symmetry group of the (numerically computed) least-perimeter cluster of n nonoverlapping circles.

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A133491 Order of the symmetry group of the (in some cases conjectural) minimal-energy configuration of n identical charged particles confined to the surface of a sphere.

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A116474 Equal divisions of the octave with progressively increasing consistency levels.

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