A055684 -id:A055684 - cp's OEIS frontend

A128113 Number of uniform polyhedra with n edges.

0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 2, 1, 0, 2, 0, 3, 3, 0, 0, 6, 0, 0, 3, 4, 0, 8, 0, 3, 5, 0, 0, 9, 0, 0, 6, 2, 0, 3, 0, 7, 4, 0, 0, 13, 0, 0, 8, 8, 0, 3, 0, 4, 9, 0, 0, 22, 0, 0, 6, 5, 0, 5, 0, 11, 11, 0, 0, 11, 0, 0, 10, 12, 0, 6, 0, 6, 9, 0, 0, 14, 0, 0, 14, 6, 0, 10, 0, 15, 15, 0, 0, 13, 0, 0

Offset: 1

Paulo de A. Sachs (sachs6(AT)yahoo.de), Feb 15 2007, corrected Feb 15 2007

nonn

			The first nonzero term, a(6)=1, represents the polyhedron with least edges: the tetrahedron. There is no polyhedron with 7 edges and no polyhedron with 8 edges is uniform, a(9)=1 represents the triangular prism, the next nonzero term, a(12), is 3 because there are the tetrahemihexahedron, the cube and the octahedron.

Hart, George W., Uniform Polyhedra.
Maeder, Roman E., Uniform Polyhedra.
Eric Weisstein's World of Mathematics, Uniform Polyhedron.

Cf. A128112, A128114.

After 240th term, a(n) equals the sum between [A055684(n/3) + 1 for n != 0 mod 3, otherwise 0] and [A055684(n/4) + A128115(n/4) + 1 for n != 0 mod 4, otherwise 0].

A339002 Numbers of the form prime(x) * prime(y) where x and y are distinct and have a common divisor > 1.

Original entry on oeis.org

21, 39, 57, 65, 87, 91, 111, 115, 129, 133, 159, 183, 185, 203, 213, 235, 237, 247, 259, 267, 299, 301, 303, 305, 319, 321, 339, 365, 371, 377, 393, 417, 427, 445, 453, 481, 489, 497, 515, 517, 519, 543, 551, 553, 559, 565, 579, 597, 611, 623, 669, 685, 687

Offset: 1

Gus Wiseman, Nov 22 2020

nonn

			The sequence of terms together with their prime indices begins:
     21: {2,4}     235: {3,15}    393: {2,32}
     39: {2,6}     237: {2,22}    417: {2,34}
     57: {2,8}     247: {6,8}     427: {4,18}
     65: {3,6}     259: {4,12}    445: {3,24}
     87: {2,10}    267: {2,24}    453: {2,36}
     91: {4,6}     299: {6,9}     481: {6,12}
    111: {2,12}    301: {4,14}    489: {2,38}
    115: {3,9}     303: {2,26}    497: {4,20}
    129: {2,14}    305: {3,18}    515: {3,27}
    133: {4,8}     319: {5,10}    517: {5,15}
    159: {2,16}    321: {2,28}    519: {2,40}
    183: {2,18}    339: {2,30}    543: {2,42}
    185: {3,12}    365: {3,21}    551: {8,10}
    203: {4,10}    371: {4,16}    553: {4,22}
    213: {2,20}    377: {6,10}    559: {6,14}

A300912 is the complement in A001358.
A338909 is the not necessarily squarefree version.
A001358 lists semiprimes, with odd and even terms A046315 and A100484.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A339005 lists products of pairs of distinct primes of divisible index.
A320656 counts factorizations into squarefree semiprimes.
A338898, A338912, and A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
A338899, A270650, and A270652 give the prime indices of squarefree semiprimes, with difference A338900.
A338910/A338911 list products of pairs of primes both of odd/even index.
A339003/A339004 list squarefree semiprimes of odd/even index.
Cf. A001221, A001222, A055684, A056239, A112798, A115392, A166237, A167171, A318990, A320892, A320911, A338901.

Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&&GCD@@PrimePi/@First/@FactorInteger[#]>1&]

A128115 Mobius inversion of A103221.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 2, 1, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 3, 2, 3, 2, 5, 2, 5, 3, 3, 2, 4, 2, 6, 3, 4, 2, 7, 2, 7, 4, 4, 3, 8, 3, 7, 4, 5, 4, 9, 3, 6, 4, 6, 4, 10, 2, 10, 5, 6, 5, 8, 4, 11, 6, 7, 4, 12, 4, 12, 6, 7, 6, 10, 4, 13, 6, 9, 6, 14, 4, 10, 7, 9, 6, 15, 4, 12, 8, 10, 7, 12, 5, 16, 7

Offset: 1

Paulo de Almeida Sachs (sachs6(AT)yahoo.de), Feb 15 2007

nonn

Number of uniform n-grammic crossed antiprisms.
Agrees with Mobius inversion of A008615 for n != 3. - Andrew Baxter, Jun 06 2008
Number of primitive equivalence classes of period 2n billiards on an equilateral triangle. - Andrew Baxter, Jun 06 2008

Andrew M. Baxter and Ron Umble, Periodic Orbits of Billiards on an Equilateral Triangle, Amer. Math. Monthly, 115 (No. 6, 2008), 479-491.

Cf. A055684.

Cf. A008615, A103221.

with(numtheory): A103221:=n->floor((n+2)/2)-floor((n+2)/3): A128115:=n->add(mobius(d)*A103221(n/d), d in divisors(n)): # Andrew Baxter, Jun 06 2008

SUM_{d|n} mu(d) * A103221(n/d), where mu is Mobius function (A008683). - Andrew Baxter, Jun 06 2008

Edited by Andrew Baxter, Jun 06 2008

A102302 Largest number < n/2 coprime to n.

Original entry on oeis.org

3, 3, 4, 3, 5, 5, 6, 5, 7, 7, 8, 7, 9, 9, 10, 9, 11, 11, 12, 11, 13, 13, 14, 13, 15, 15, 16, 15, 17, 17, 18, 17, 19, 19, 20, 19, 21, 21, 22, 21, 23, 23, 24, 23, 25, 25, 26, 25, 27, 27, 28, 27, 29, 29, 30, 29, 31, 31, 32, 31, 33, 33, 34, 33, 35, 35, 36, 35, 37, 37, 38, 37, 39, 39

Offset: 7

Hugo Pfoertner, Jan 23 2005

The densest possible star-shaped regular n-gon is formed by connecting with straight lines every a(n)-th point out of n regularly spaced points lying on a circumference.

For a given n there are A055684(n) different star-shaped regular polygons. The minimum skip increment for connecting points on the circumference is given by A053669(n), the maximum skip increment is given by a(n). There are no star-shaped polygons for n=3,4,6 and unique star-shaped polygons for n=5,8,10 and 12, for which a(n) = A053669(n).

Colin Barker, Table of n, a(n) for n = 7..1000
Jay Kappraff, Gary W. Adamson, Polygons and Chaos, BRIDGES Mathematical Connections in Art, Music, and Science, 2001.
Hugo Pfoertner, Star-shaped regular polygons.
Hugo Pfoertner, (Star-Shaped-) Polygons with Maximal Density.
Eric Weisstein's World of Mathematics, Star Polygon.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A053669 (least number coprime to n), A055684 (number of different n-pointed stars).

lnc[n_]:=Module[{k=Floor[n/2]},While[!CoprimeQ[n,k],k--];k]; Array[ lnc,90,7] (* Harvey P. Dale, May 15 2021 *)

Vec(-x^7*(x^4+x^3-x^2-3)/((x-1)^2*(x+1)*(x^2+1)) + O(x^100)) \\ Colin Barker, Feb 21 2015

a(4*k-1) = a(4*k) = a(4*k+2) = 2*k-1; a(4*k+1) = 2*k.
a(n) = (1/2) (n - (I^n + (-I)^n)/2 - (-1)^n + 4). - Ralf Stephan, May 17 2007
a(n) = a(n-1)+a(n-4)-a(n-5) for n>11. - Colin Barker, Feb 21 2015
G.f.: -x^7*(x^4+x^3-x^2-3) / ((x-1)^2*(x+1)*(x^2+1)). - Colin Barker, Feb 21 2015

A338333 Number of relatively prime 3-part strict integer partitions of n with no 1's.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 4, 7, 6, 10, 8, 14, 12, 18, 16, 24, 18, 30, 25, 34, 30, 44, 31, 52, 42, 56, 49, 69, 50, 80, 64, 83, 70, 102, 71, 114, 90, 112, 100, 140, 98, 153, 117, 153, 132, 184, 128, 195, 154, 196, 169, 234, 156, 252, 196, 241

Offset: 0

Gus Wiseman, Oct 30 2020

nonn

The Heinz numbers of these partitions are the intersection of A005117 (strict), A005408 (no 1's), A014612 (length 3), and A289509 (relatively prime).

			The a(9) = 1 through a(19) = 14 triples (A = 10, B = 11, C = 12, D = 13, E = 14):
  432   532   542   543   643   653   654   754   764   765   865
              632   732   652   743   753   763   854   873   874
                          742   752   762   853   863   954   964
                          832   932   843   943   872   972   973
                                      852   952   953   A53   982
                                      942   B32   962   B43   A54
                                      A32         A43   B52   A63
                                                  A52   D32   A72
                                                  B42         B53
                                                  C32         B62
                                                              C43
                                                              C52
                                                              D42
                                                              E32

A001399(n-9) does not require relative primality.
A005117 /\ A005408 /\ A014612 /\ A289509 gives the Heinz numbers.
A055684 is the 2-part version.
A284825 counts the case that is also pairwise non-coprime.
A337452 counts these partitions of any length.
A337563 is the pairwise coprime instead of relatively prime version.
A337605 is the pairwise non-coprime instead of relative prime version.
A338332 is the not necessarily strict version.
A338333*6 is the ordered version.
A000837 counts relatively prime partitions.
A008284 counts partitions by sum and length.
A078374 counts relatively prime strict partitions.
A101271 counts 3-part relatively prime strict partitions.
A220377 counts 3-part pairwise coprime strict partitions.
A337601 counts 3-part partitions whose distinct parts are pairwise coprime.
Cf. A000010, A000217, A000741, A023022, A082024, A302698, A307719, A337450, A337599, A338468.

Table[Length[Select[IntegerPartitions[n,{3}],UnsameQ@@#&&!MemberQ[#,1]&&GCD@@#==1&]],{n,0,30}]

A370459 Number of unicursal stars with n vertices.

Original entry on oeis.org

0, 0, 1, 1, 5, 19, 112, 828, 7441, 76579, 871225, 10809051, 144730446, 2079635889, 31912025537, 520913578812, 9013780062785, 164829273635749, 3176388519597555, 64343477504391475, 1366925655386979893, 30390554390984325019, 705740995420852895453

Offset: 3

Adam M. Scherlis, Feb 19 2024

nonn

A unicursal star is a closed loop formed by diagonals of a regular n-gon.
These are Hamiltonian cycles on the graph complement of the n-cycle.
Allowing polygon diagonals, but not sides, is equivalent to requiring every edge to cross at least one other edge.
These are counted up to rotation and reflection, i.e., modulo dihedral symmetry of the n-gon.
Inspired by a unicursal dodecagram drawn by Gordon FitzGerald (see links).

			For n=5, there is only the regular pentagram {5/2}.
For n=6, there is only the unicursal hexagram.
For n=7, in addition to the two regular heptagrams {7/2} and {7/3}, there are three nontrivial unicursal heptagrams represented by:
 (0, 2, 4, 1, 6, 3, 5, 0)
 (0, 2, 5, 1, 3, 6, 4, 0)
 (0, 2, 5, 1, 4, 6, 3, 0).

Andrew Howroyd, Table of n, a(n) for n = 3..200
Gordon FitzGerald, illustration of a unicursal dodecagram.
Wikipedia, Unicursal hexagram.

Cf. A000940 (polygon sides allowed).
Cf. A055684 (cases with dihedral symmetry only).
Cf. A002816 (rotations and reflections counted separately).
Cf. A231091 (up to rotations only), A370769 (achiral).

\\ Requires a370068 from A370068.
Ro(n)=-(-1)^n + subst(serlaplace(polcoef(((1 - x)^2)/(2*(1 + x)*(1 + (1 - 2*y)*x + 2*y*x^2)) + O(x*x^n), n)), y, 1)
Re(n)=subst(serlaplace(polcoef((1 - x - 2*x^2)/(4*(1 + (1 - 2*y)*x + 2*y*x^2)) + O(x*x^n), n)), y, 1)
a(n)={if(n<3, 0, (if(n%2, 2*Ro(n\2), Re(n/2)) + a370068(n))/4)} \\ Andrew Howroyd, Mar 01 2024

a(n) = (A231091(n) + A370769(n))/2. - Andrew Howroyd, Mar 06 2024

a(14) onwards from Andrew Howroyd, Feb 26 2024

A330662 Triangle read by rows: T(n,k) is the number of polygons with 2n sides, of which k run through the center of a circle, on the circumference of which the 2n vertices of the polygon are arranged at equal spacing.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 16, 24, 12, 8, 744, 960, 576, 192, 48, 56256, 69120, 39360, 13440, 2880, 384, 6385920, 7580160, 4204800, 1420800, 316800, 46080, 3840, 1018114560, 1178956800, 642539520, 216115200, 49190400, 7741440, 806400, 46080

Offset: 0

Ludovic Schwob, Dec 23 2019

Rotations and reflections are counted separately.
By "2*n-sided polygons" we mean the polygons that can be drawn by connecting 2*n equally spaced points on a circle.
T(0,0)=0 and T(0,1)=1 by convention.
The sequence is limited to even-sided polygons, since all odd-sided polygons have no side passing through the center.

			Triangle begins:
    0;
    0,   1;
    1,   0,   2;
   16,  24,  12,   8;
  744, 960, 576, 192, 48;

Ludovic Schwob, Table of n, a(n) for n = 0..494
Ludovic Schwob, Illustration of T(3,k), 0≤k≤3.

Row sums give A001710(2*n-1) (number of polygons with 2*n sides).
Cf. A000165 (diagonal).
Star polygons: A014106, A055684, A102302.
Cf. A309318.

T := (n, k) -> `if`(n<2, k, 2^(k-1)*binomial(n,k)*(2*n-k-1)!*hypergeom([k-n], [k-2*n+ 1], -2)):
seq(seq(simplify(T(n,k)), k=0..n),n=0..7); # Peter Luschny, Jan 07 2020

T(n,n) = 2^(n-1) * (n-1)! for all n >= 1.
T(n,0) = A307923(n) for all n>=1.
T(n,k) = binomial(n,k)* Sum_{i=k..n} (-1)^(i-k)*binomial(n-k,i-k)*(2n-1-i)!*2^(i-1), for n>=2 and 0<=k<=n.

A338332 Number of relatively prime 3-part integer partitions of n with no 1's.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 5, 3, 8, 6, 9, 9, 16, 10, 21, 15, 22, 20, 33, 21, 38, 30, 41, 35, 56, 34, 65, 49, 64, 56, 79, 55, 96, 72, 93, 77, 120, 76, 133, 99, 122, 110, 161, 105, 172, 126, 167, 143, 208, 136, 213, 165, 212, 182, 261, 163, 280, 210, 257

Offset: 0

Gus Wiseman, Oct 30 2020

nonn

The Heinz numbers of these partitions are the intersection of A005408 (no 1's), A014612 (length 3), and A289509 (relatively prime).

			The a(7) = 1 through a(17) = 16 triples (A = 10, B = 11, C = 12, D = 13):
  322   332   432   433   443   543   544   554   654   655   665
              522   532   533   552   553   653   744   754   755
                          542   732   643   743   753   763   764
                          632         652   752   762   772   773
                          722         733   833   843   853   854
                                      742   932   852   943   863
                                      832         942   952   872
                                      922         A32   A33   944
                                                  B22   B32   953
                                                              962
                                                              A43
                                                              A52
                                                              B33
                                                              B42
                                                              C32
                                                              D22

A001399(n-6) does not require relative primality.
A005408 /\ A014612 /\ A289509 gives the Heinz numbers of these partitions.
A055684 is the 2-part version.
A284825 counts the case that is also pairwise non-coprime.
A302698 counts these partitions of any length.
A337563 is the pairwise coprime instead of relatively prime version.
A338333 is the strict version.
A000837 counts relatively prime partitions, with strict case A078374.
A008284 counts partitions by sum and length.
Cf. A000010, A000741, A023022, A078374, A082024, A101271, A307719, A337450, A337599, A337600, A337601.

Table[Length[Select[IntegerPartitions[n,{3}],!MemberQ[#,1]&&GCD@@#==1&]],{n,0,30}]

A338468 Odd squarefree numbers whose prime indices have no common divisor > 1.

Original entry on oeis.org

15, 33, 35, 51, 55, 69, 77, 85, 93, 95, 105, 119, 123, 141, 143, 145, 155, 161, 165, 177, 187, 195, 201, 205, 209, 215, 217, 219, 221, 231, 249, 253, 255, 265, 285, 287, 291, 295, 309, 323, 327, 329, 335, 341, 345, 355, 357, 381, 385, 391, 395, 403, 407, 411

Offset: 1

Gus Wiseman, Oct 29 2020

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Also Heinz numbers of relatively prime strict integer partitions with no 1's (A337452). The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
     15: {2,3}      145: {3,10}     249: {2,23}     355: {3,20}
     33: {2,5}      155: {3,11}     253: {5,9}      357: {2,4,7}
     35: {3,4}      161: {4,9}      255: {2,3,7}    381: {2,31}
     51: {2,7}      165: {2,3,5}    265: {3,16}     385: {3,4,5}
     55: {3,5}      177: {2,17}     285: {2,3,8}    391: {7,9}
     69: {2,9}      187: {5,7}      287: {4,13}     395: {3,22}
     77: {4,5}      195: {2,3,6}    291: {2,25}     403: {6,11}
     85: {3,7}      201: {2,19}     295: {3,17}     407: {5,12}
     93: {2,11}     205: {3,13}     309: {2,27}     411: {2,33}
     95: {3,8}      209: {5,8}      323: {7,8}      413: {4,17}
    105: {2,3,4}    215: {3,14}     327: {2,29}     415: {3,23}
    119: {4,7}      217: {4,11}     329: {4,15}     429: {2,5,6}
    123: {2,13}     219: {2,21}     335: {3,19}     435: {2,3,10}
    141: {2,15}     221: {6,7}      341: {5,11}     437: {8,9}
    143: {5,6}      231: {2,4,5}    345: {2,3,9}    447: {2,35}

A302568 is the prime or pairwise coprime version, counted by A007359.
A302697 is not required to be squarefree, counted by A302698 (ordered version: A337450).
A302796 allows evens, counted by A078374 (ordered version: A332004).
A337452 counts partitions with these Heinz numbers (ordered version: A337451).
A337984 is the pairwise coprime version, counted by A337485 (ordered version: A337697).
A005117 lists squarefree numbers.
A005408 lists odd numbers.
A056911 lists odd squarefree numbers.
A289509 lists Heinz numbers of relatively prime partitions, counted by A000837 (ordered version: A000740).
Cf. A000010, A007359, A051424, A055684, A056239, A101268, A289508, A302505, A302569, A302696, A302798, A337694.

Select[Range[1,100,2],SquareFreeQ[#]&&GCD@@PrimePi/@First/@FactorInteger[#]==1&]

A065802 How small is the squeezed n-gon? Let s0 be the side of a regular n-gon and s1 the side of the maximal n-gon which can be squeezed between the former and its circumcircle. The n-th entry in the sequence is floor(s0/s1).

Original entry on oeis.org

3, 5, 9, 13, 19, 24, 32, 38, 48, 56, 67, 77, 90, 102, 116, 129, 145, 160, 178, 194, 213, 231, 252, 272, 294, 316, 340, 363, 388, 413, 440, 466, 495, 523, 554, 583, 615, 646, 680, 713, 748, 782, 820, 855, 894, 932, 972, 1011, 1053, 1094, 1137, 1180, 1225

Offset: 3

Rainer Rosenthal, Dec 05 2001

Closely related to K(n) = (2*n/Pi)*sin(Pi/n)/(1-cos(Pi/n)) as derived from the n-gon with same circumference as the circle squeezed between the large n-gon and its circumcircle.

			a(3) = 3 as can be seen in Christmas stars: cos(Pi/3)=1/2, thus a(3) = floor((3/2)/(1/2)) = 3. a(4) = 5 as proposed by Bill Taylor in sci.math: tan(Pi/4)=1, thus a(4) = floor(2*(2/1^2) + 1) = 5.

Robert Israel, Table of n, a(n) for n = 3..10000
Bill Taylor, Ignacio Larrosa Cañestro, Rainer Rosenthal, Little Geometry Problem, thread in newsgroup sci.math, Oct 31 - Nov 06, 2001.

Cf. A055684.

f:= proc(n) if n::odd then floor((1+cos(Pi/n))/(1-cos(Pi/n))) else floor(2*(2/(tan(Pi/n))^2) + 1) fi end proc:
map(f, [$3..100]); # Robert Israel, Oct 24 2017

f[n_] := If[ OddQ[n], Floor[(1 + Cos[Pi/n]) / (1 - Cos[Pi/n])], Floor[4/(Tan[Pi/n])^2 + 1] ]; Table[ f[n], {n, 3, 60} ]

For n=odd: a(n) = floor((1+cos(Pi/n))/(1-cos(Pi/n))) For n=even: a(n) = floor( 2*(2/(tan(Pi/n))^2) + 1 )

a(n) = floor(4*n^2/Pi^2) - b(n) where b(n) is in {0,1,2}; 0 occurs only for odd n, while 2 occurs only for even n. - Robert Israel, Oct 24 2017

More terms from Robert G. Wilson v, Dec 06 2001

cp's OEIS Frontend

A128113 Number of uniform polyhedra with n edges.

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A339002 Numbers of the form prime(x) * prime(y) where x and y are distinct and have a common divisor > 1.

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A128115 Mobius inversion of A103221.

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A102302 Largest number < n/2 coprime to n.

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A338333 Number of relatively prime 3-part strict integer partitions of n with no 1's.

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A370459 Number of unicursal stars with n vertices.

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A330662 Triangle read by rows: T(n,k) is the number of polygons with 2*n sides, of which k run through the center of a circle, on the circumference of which the 2*n vertices of the polygon are arranged at equal spacing.

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A338332 Number of relatively prime 3-part integer partitions of n with no 1's.

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A330662 Triangle read by rows: T(n,k) is the number of polygons with 2n sides, of which k run through the center of a circle, on the circumference of which the 2n vertices of the polygon are arranged at equal spacing.