James Propp - cp's OEIS frontend

A382529 The composite numbers ordered by decreasing reciprocal-distance from the primes (reciprocal distance defined in comments).

4, 6, 9, 8, 10, 15, 12, 14, 21, 26, 16, 25, 18, 20, 27, 34, 22, 33, 24, 35, 39, 28, 50, 30, 32, 45, 56, 49, 36, 51, 38, 64, 55, 40, 57, 42, 44, 63, 76, 46, 120, 65, 119, 93, 48, 69, 86, 121, 118, 92, 75, 52, 54, 94, 77, 122, 117, 81, 58, 85, 60, 62, 87, 123, 91, 144

Offset: 1

James Propp, Mar 30 2025

nonn

The reciprocal distance between m and p is defined as d(m,p) = abs(1/m - 1/p).
The distance between a composite number m and the set of primes is d(m) = Min_{p prime} d(m,p), which means considering p which is the next prime below m, and q the next prime above m.
Bertrand's postulate p > m/2 means d(m) < 1/m so that all m with d(m) > epsilon are m < 1/epsilon.
The plot (e.g., ListPlot in Mathematica) shows interesting large-scale structure.

			a(3) and a(4) are 9 and 8 respectively. 9 precedes 8 in the list of composites because min(1/7-1/9,1/9-1/11) is greater than min(1/7-1/8,1/8-1/11).

James Propp, Table of n, a(n) for n = 1..1440
Hugo Pfoertner, Plot of a(n) and 4*n for n<=250000.
James Propp, Plot of a(n) for n = 1..1440

Cf. A000040, A002386, A002808, A005250, A354604.

epsilon = .0005; (* terms < 1/epsilon *);
table = {}; For[m = 2, m <= 1/epsilon, m++, If[! PrimeQ[m], i = m; While[! PrimeQ[i], --i]; j = m; While[! PrimeQ[j], ++j]; dist = Min[1/i - 1/m, 1/m - 1/j]; If[dist > epsilon, table = Append[table, {dist, m}]]]]; init = Reverse[Sort[table]]; Transpose[init][[2]]

a382529(nterms) = {my(m=nterms+nterms/log(nterms)+3*nterms/log(nterms)^2, mc=floor(m*if(nterms<337963, 4, log(m)/2-1)), C=vectorsmall(mc), L=List(), nc=0); forcomposite(n=4, mc, C[nc++]=n; my(d=min(1/precprime(n)-1/n,1/n-1/nextprime(n))); listput(L,d)); my(P=vecsort(L,,5)); vecextract(Vec(C),P)[1..nterms]}; \\ Hugo Pfoertner, Apr 22 2025

More terms from Hugo Pfoertner, Mar 30 2025

A377309 Number of stones-and-bones tilings of an n-triangle.

Original entry on oeis.org

1, 0, 1, 3, 0, 30, 246, 0, 25321, 591103, 0, 603105309, 41333676318, 0, 410382321560202, 83918368144461643, 0, 8025244898075570226296, 4941312847984149589980261

Offset: 0

James Propp, Oct 23 2024

These numbers were computed using the TilingCount program of David Desjardins. Stones are also called T_2 tiles and bones are also called L_3 tiles.

			For n=3 the three tilings are
      A
     A A
    B B B
and
      A
     A B
    A B B
and
      A
     B A
    B B A

James Propp, The Benzel Tilings Site

Cf. A334875.

a(n)=0 when n is congruent to 1 mod 3, since in that case the number of cells in the n-triangle is not divisible by 3.

A364483 The number of stones-and-bones tilings of the (3n-1,3n)-benzel when one orientation of bone is forbidden.

Original entry on oeis.org

1, 16, 3360, 9371648, 347950546944, 172066422921363456, 1133503548832944876421120

Offset: 1

James Propp, Jul 26 2023

See Problem 11 of the Propp article. It is conjectured that a(n) is 2-adically continuous as a function of n.

James Propp, Trimer covers in the triangular grid: twenty mostly open problems, arXiv:2206.06472 [math.CO], 2022.

Cf. A352207, A364481, A364482, A364416, A364417, A364418, A364438.

A364482 The number of stones-and-bones tilings of the (3n+1,3n+2)-benzel when one orientation of bone is forbidden.

Original entry on oeis.org

6, 512, 591360, 9160359936, 1897011087409152, 5244422625774526267392, 193403358706333224417833779200, 95098462720808932931887549372170240000

Offset: 1

James Propp, Jul 26 2023

nonn

See Problem 10 of the Propp article. It is conjectured that a(n) is 2-adically continuous as a function of n.

James Propp, Trimer covers in the triangular grid: twenty mostly open problems, arXiv:2206.06472 [math.CO], 2022.

Cf. A352207, A364481, A364483, A364416, A364417, A364418, A364438.

A364481 The number of stones-and-bones tilings of the (3n+1,3n+1)-benzel when one orientation of bone is forbidden.

Original entry on oeis.org

4, 224, 168960, 1705639936, 229940737867776, 413561647491497066496, 9918120959299139713735065600

Offset: 1

James Propp, Jul 26 2023

See Problem 9 of the Propp article. It is conjectured that a(n) is 2-adically continuous as a function of n.

James Propp, Trimer covers in the triangular grid: twenty mostly open problems, arXiv:2206.06472 [math.CO], 2022.

Cf. A352207, A364482, A364483, A364416, A364417, A364418, A364438.

A364438 The sum of the weights of the stones-and-bones tilings of the (n,2n-3)-benzel, where each stone is given multiplicative weight 3.

Original entry on oeis.org

9, 270, 27110, 8798490, 8980383330, 28344705113430, 273927748387623390, 8057418594145673168610, 718650987298253553656580570, 193874673319110717570773876192670, 157927323459469084048485672225266775510, 387962431958247267773527802272080627127318890

Offset: 3

James Propp, Jul 24 2023

nonn

See Problem 18 of the Propp article. It is conjectured that a(n) is 2-adically continuous as a function of n > 3.

James Propp, Table of n, a(n) for n = 3..17
James Propp, Trimer covers in the triangular grid: twenty mostly open problems, arXiv:2206.06472 [math.CO], 2022.

Cf. A352207, A364416, A364417, A364418, A364481, A364482, A364483.

A364418 The sum of the weights of the stones-and-bones tilings of the (n,2n-4)-benzel, where each stone is given multiplicative weight 3.

Original entry on oeis.org

3, 102, 10260, 3267540, 3272495580, 10170919805580, 97112573496153540, 2829427113881208115260, 250440846963119234063024220, 67143197168392738521628168122420, 54411613647618445838464808052508179060

Offset: 3

James Propp, Jul 23 2023

nonn

See Problem 17 of the Propp article. It is conjectured that a(n) is 2-adically continuous as a function of n > 4.

J. Propp, Trimer covers in the triangular grid: twenty mostly open problems, to appear.

James Propp, Table of n, a(n) for n = 3..17
James Propp, Trimer covers in the triangular grid: twenty mostly open problems, arXiv:2206.06472 [math.CO], 2022.

Cf. A352207, A364416, A364417, A364438, A364481, A364482, A364483.

A364417 The number of stones-and-bones tilings of the (n,2n-3)-benzel.

Original entry on oeis.org

3, 18, 142, 1266, 12030, 118650, 1198230, 12296202, 127633590, 1336133730, 14079114270, 149124688482, 1586159072814, 16929780310218, 181227223899942, 1944808008842490, 20915277691567206

Offset: 3

James Propp, Jul 23 2023

See Problem 16 of the Propp article. It is conjectured that a(n) is 2-adically continuous as a function of n > 3.

James Propp, Trimer covers in the triangular grid: twenty mostly open problems, arXiv:2206.06472 [math.CO], 2022.

Cf. A352207, A364416, A364418, A364438, A364481, A364482, A364483.

a(15)-a(19) from James Propp, Apr 25 2024

A364416 The number of stones-and-bones tilings of the (n,2n-4)-benzel.

Original entry on oeis.org

1, 10, 84, 724, 6516, 60900, 586404, 5777916, 57952212, 589381020, 6060195316, 62863155972, 656765033268, 6902094928308, 72892778268996, 773013952508268, 8226672021670804, 87817595880192684

Offset: 3

James Propp, Jul 23 2023

See Problem 15 of the Propp article. It is conjectured that a(n) is 2-adically continuous as a function of n>4.

James Propp, Trimer covers in the triangular grid: twenty mostly open problems, arXiv:2206.06472 [math.CO], 2022.

Cf. A352207, A364417, A364418, A364438, A364481, A364482, A364483.

More terms from James Propp, Apr 26 2024

A364134 Number of tilings of a (k(3k-1)/2, k(3k+1)/2)-benzel by bones.

Original entry on oeis.org

1, 2, 42705, 7501790059160666750

Offset: 1

James Propp, Jul 22 2023

			For k=2 the two tilings are shown in Figure 18 of the (J. Kim and J. Propp) arxiv.org link.

J. Kim and J. Propp, A pentagonal number theorem for tribone tilings, Electronic Journal of Combinatorics, to appear.

J. Kim and J. Propp, A pentagonal number theorem for tribune tilings, arXiv preprint, arXiv:2206.04223 [math.CO], 2022-2023.

cp's OEIS Frontend

User: James Propp

A382529 The composite numbers ordered by decreasing reciprocal-distance from the primes (reciprocal distance defined in comments).

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A377309 Number of stones-and-bones tilings of an n-triangle.

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A364483 The number of stones-and-bones tilings of the (3n-1,3n)-benzel when one orientation of bone is forbidden.

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A364482 The number of stones-and-bones tilings of the (3n+1,3n+2)-benzel when one orientation of bone is forbidden.

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A364481 The number of stones-and-bones tilings of the (3n+1,3n+1)-benzel when one orientation of bone is forbidden.

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A364438 The sum of the weights of the stones-and-bones tilings of the (n,2n-3)-benzel, where each stone is given multiplicative weight 3.

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A364418 The sum of the weights of the stones-and-bones tilings of the (n,2n-4)-benzel, where each stone is given multiplicative weight 3.

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A364417 The number of stones-and-bones tilings of the (n,2n-3)-benzel.

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A364416 The number of stones-and-bones tilings of the (n,2n-4)-benzel.

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A364134 Number of tilings of a (k*(3k-1)/2, k*(3k+1)/2)-benzel by bones.

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A364134 Number of tilings of a (k(3k-1)/2, k(3k+1)/2)-benzel by bones.