Pontus von Brömssen - cp's OEIS frontend

A387005 Number of free (d,2)-polyominoids of size n in arbitrary dimension d.

1, 2, 12, 103

Offset: 1

Pontus von Brömssen, Aug 14 2025

Wikipedia, Polyominoid.
Index entries for sequences related to polyominoes.

Main diagonal of A387003.
Row sums of A387004.
Cf. A005519 (polyominoes), A365565 (polysticks).
Cf. A000105 (2 dimensions), A075679 (3 dimensions), A366334 (4 dimensions).

A387002 Triangle read by rows: T(n,d) is the number of fixed, properly d-dimensional (d,2)-polyominoids of size n, 2 <= d <= n+1.

Original entry on oeis.org

1, 2, 12, 6, 140, 320, 19, 1554, 10368, 13520, 63, 17622, 265344, 892864, 786432, 216, 206747, 6390484, 41998840, 89389920, 58383808, 760, 2503578, 152166240, 1749529040, 6773387520

Offset: 1

Pontus von Brömssen, Aug 14 2025

A (d,2)-polyominoid consists of unit square cells with integer coordinates in the d-dimensional grid, where two cells are connected if they share an edge. The polyominoid is properly d-dimensional if it is not contained in a (d-1)-dimensional affine subspace.

			Triangle begins:
  n\d |     2          3           4           5          6        7  8  9 10 11
  ----+-------------------------------------------------------------------------
   1  |     1
   2  |     2         12
   3  |     6        140         320
   4  |    19       1554       10368       13520
   5  |    63      17622      265344      892864     786432
   6  |   216     206747     6390484    41998840   89389920 58383808
   7  |   760    2503578   152166240  1749529040 6773387520        ?  ?
   8  |  2725   31117536  3644734836 69246650605          ?        ?  ?  ?
   9  |  9910  394953243 88344741448           ?          ?        ?  ?  ?  ?
  10  | 36446 5098388985           ?           ?          ?        ?  ?  ?  ?  ?

Wikipedia, Polyominoid.
Index entries for sequences related to polyominoes.

Cf. A001168 (column d=2), A195739 (polyominoes), A385582 (polysticks), A385715, A387004 (free).

T(n,d) = Sum_{k=2..d} (-1)^(d-k)*binomial(d,k)*A385715(k,n), i.e., the n-th row is the inverse binomial transform of the n-th column of A385715 (with the convention that T(n,d) = A385715(d,n) = 0 when d <= 1).

A387004 Triangle read by rows: T(n,d) is the number of free, properly d-dimensional (d,2)-polyominoids of size n, 2 <= d <= n+1.

Original entry on oeis.org

1, 1, 1, 2, 7, 3, 5, 49, 41, 8

Offset: 1

Pontus von Brömssen, Aug 14 2025

			Triangle begins:
  n\d| 2  3  4  5
  ---+-----------
  1  | 1
  2  | 1  1
  3  | 2  7  3
  4  | 5 49 41  8

Wikipedia, Polyominoid.
Index entries for sequences related to polyominoes.

Cf. A000105 (column d=2), A049430 (polyominoes), A365566 (polysticks), A387002 (fixed), A387003, A387005 (row sums).

T(n,d) = A387003(n,d) - A387003(n,d-1) (with A387003(n,1) = 0).

A387003 Triangle read by rows: T(n,d) is the number of free (d,2)-polyominoids of size n, 2 <= d <= n+1.

Original entry on oeis.org

1, 1, 2, 2, 9, 12, 5, 54, 95, 103

Offset: 1

Pontus von Brömssen, Aug 14 2025

If d > n+1, there are T(n,n+1) such polyominoids. The triangle only includes the values for d <= n+1.

			Triangle begins:
  n\d| 2  3  4   5
  ---+------------
  1  | 1
  2  | 1  2
  3  | 2  9 12
  4  | 5 54 95 103

Wikipedia, Polyominoid.
Index entries for sequences related to polyominoes.

Columns: A000105 (d=2), A075679 (d=3), A366334 (d=4).

Cf. A330891 (polyominoes), A385583 (polysticks), A385715 (fixed), A387002, A387004, A387005 (main diagonal).

T(n,d) = Sum_{k=1..d} A387004(n,k).

A386007 Least k such that there are exactly n primes that are popular on the interval [2,k] (see A385503); i.e., exactly n primes share the lead as the most common greatest prime factor of the numbers 2..k.

Original entry on oeis.org

2, 3, 70, 2626355

Offset: 1

Pontus von Brömssen, Jul 14 2025

			a(3) = 70, because the 3 primes 3, 5, and 7 all occur A385652(70) = 10 times (the maximum) as the greatest prime factor of the numbers 2..70, and for earlier intervals there is never a tie between 3 numbers.
a(4) = 2626355, because the 4 primes 73, 83, 109, and 113 all occur A385652(2626355) = 7634 times (the maximum) as the greatest prime factor of the numbers 2..2626355, and for earlier intervals there is never a tie between 4 numbers.

Cf. A006530, A385503, A385652.

gpf[n_]:=FactorInteger[n][[-1,1]];a[n_]:=Module[{k=1},Until[Length[Commonest[gpf/@Range[2,k]]]==n,k++];k] (* James C. McMahon, Jul 20 2025 *)

A385654 Greatest prime factor of A385653(n).

Original entry on oeis.org

2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 13, 13, 13, 13, 13, 7, 13, 13, 13, 13, 13, 13

Offset: 1

Pontus von Brömssen, Jul 06 2025

nonn

A number appears in this sequence if and only if it is a uniquely popular prime; see A289662. a(n) is uniquely popular on the interval [2,A385653(n)].

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A006530, A289662, A385503, A385652, A385653.

gpf(n) = if (n==1,1, vecmax(factor(n)[,1])); \\ A006530
f(n) = my(v=vector(n, k, gpf(k)), s=Set(v)); vecmax(apply(x->#x, vector(#s, i, select(x->(x==s[i]), v)))); \\ A385652
a(n) = my(k=2); while (f(k) !=n, k++); gpf(k); \\ Michel Marcus, Jul 06 2025

A385653 Least k such that A385652(k) = n.

Original entry on oeis.org

2, 4, 8, 12, 18, 24, 27, 36, 48, 54, 72, 80, 90, 100, 120, 125, 135, 150, 160, 180, 196, 210, 224, 245, 252, 280, 294, 315, 336, 343, 350, 378, 392, 420, 441, 448, 490, 504, 525, 560, 567, 588, 630, 672, 686, 700, 735, 756, 784, 840, 875, 882, 896, 945, 980

Offset: 1

Pontus von Brömssen, Jul 06 2025

nonn

A385654(n) is uniquely popular on the interval [2,a(n)]; see A289662.

Equivalently, a(n) is the least k >= 2 such that A078899(k) = n.

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A006530, A078899, A289662, A385503, A385652, A385654.

gpf(n) = if (n==1,1, vecmax(factor(n)[,1])); \\ A006530
f(n) = my(v=vector(n, k, gpf(k)), s=Set(v)); vecmax(apply(x->#x, vector(#s, i, select(x->(x==s[i]), v)))); \\ A385652
a(n) = my(k=2); while (f(k) !=n, k++); k; \\ Michel Marcus, Jul 06 2025

A385652 Maximum frequency of gpf(k) for 2 <= k <= n, where gpf(k) = A006530(k) is the greatest prime factor of k.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12

Offset: 2

Pontus von Brömssen, Jul 06 2025

nonn

The prime p is popular on the interval [2,n] if it is the greatest prime factor of a(n) numbers in that interval; see A385503.

			     |     | cumulative frequencies for gpf's |
   n | gpf |    2  3  5  7 11 13 17 19 23     | a(n)
  ---+-----+----------------------------------+-----
   2 |   2 |    1  0  0  0  0  0  0  0  0     |  1
   3 |   3 |    1  1  0  0  0  0  0  0  0     |  1
   4 |   2 |    2  1  0  0  0  0  0  0  0     |  2
   5 |   5 |    2  1  1  0  0  0  0  0  0     |  2
   6 |   3 |    2  2  1  0  0  0  0  0  0     |  2
   7 |   7 |    2  2  1  1  0  0  0  0  0     |  2
   8 |   2 |    3  2  1  1  0  0  0  0  0     |  3
   9 |   3 |    3  3  1  1  0  0  0  0  0     |  3
  10 |   5 |    3  3  2  1  0  0  0  0  0     |  3
  11 |  11 |    3  3  2  1  1  0  0  0  0     |  3
  12 |   3 |    3  4  2  1  1  0  0  0  0     |  4
  13 |  13 |    3  4  2  1  1  1  0  0  0     |  4
  14 |   7 |    3  4  2  2  1  1  0  0  0     |  4
  15 |   5 |    3  4  3  2  1  1  0  0  0     |  4
  16 |   2 |    4  4  3  2  1  1  0  0  0     |  4
  17 |  17 |    4  4  3  2  1  1  1  0  0     |  4
  18 |   3 |    4  5  3  2  1  1  1  0  0     |  5
  19 |  19 |    4  5  3  2  1  1  1  1  0     |  5
  20 |   5 |    4  5  4  2  1  1  1  1  0     |  5
  21 |   7 |    4  5  4  3  1  1  1  1  0     |  5
  22 |  11 |    4  5  4  3  2  1  1  1  0     |  5
  23 |  23 |    4  5  4  3  2  1  1  1  1     |  5
  24 |   3 |    4  6  4  3  2  1  1  1  1     |  6

Pontus von Brömssen, Table of n, a(n) for n = 2..10000

Cf. A006530, A078899, A385503, A385653, A385654.

gpf(n) = if (n==1,1, vecmax(factor(n)[,1])); \\ A006530
a(n) = my(v=vector(n, k, gpf(k)), s=Set(v)); vecmax(apply(x->#x, vector(#s, i, select(x->(x==s[i]), v)))); \\ Michel Marcus, Jul 06 2025

from collections import Counter
from itertools import count
from sympy import factorint
def A385652_generator():
    c = Counter()
    M = 0
    for n in count(2):
        gpf = max(factorint(n))
        c[gpf] += 1
        if c[gpf] > M: M += 1
        yield M

a(n) = max_{k=2..n} A078899(k).

A385581 Square array read by antidiagonals: T(n,d) is the number of fixed d-dimensional polysticks of size n.

Original entry on oeis.org

1, 2, 1, 3, 6, 1, 4, 15, 22, 1, 5, 28, 95, 88, 1, 6, 45, 252, 681, 372, 1, 7, 66, 525, 2600, 5277, 1628, 1, 8, 91, 946, 7065, 29248, 43086, 7312, 1, 9, 120, 1547, 15696, 104097, 349132, 365313, 33466, 1, 10, 153, 2360, 30513, 285828, 1632915, 4351944, 3186444, 155446, 1

Offset: 1

Pontus von Brömssen, Jul 04 2025

The first 17 antidiagonals are from Mertens and Moore (2018), either directly from Table 1 or computed using the perimeter polynomials in Appendix A. T(14,5) is the only unknown value in the 18th antidiagonal.

T(13,6) = 14054816418877200 (Mertens and Moore).

			Table begins:
  n\d| 1     2       3        4         5          6          7           8
  ---+---------------------------------------------------------------------
  1  | 1     2       3        4         5          6          7           8
  2  | 1     6      15       28        45         66         91         120
  3  | 1    22      95      252       525        946       1547        2360
  4  | 1    88     681     2600      7065      15696      30513       53936
  5  | 1   372    5277    29248    104097     285828     661549     1356384
  6  | 1  1628   43086   349132   1632915    5551480   15314936    36449288
  7  | 1  7312  365313  4351944  26817465  113045832  372033993  1028383408
  8  | 1 33466 3186444 56062681 456137580 2386821009 9377038237 30118187174

Pontus von Brömssen, Table of n, a(n) for n = 1..153 (first 17 antidiagonals)
Stephan Mertens and Cristopher Moore, Series expansion of the percolation threshold on hypercubic lattices, J. Phys. A: Math. Theor., 51 (2018), 475001; arXiv:1805.02701 [cond-mat.stat-mech], 2018.
Index entries for sequences related to polyominoes.

Cf. A000384 (row n=2), A385291 (polyominoes), A385582, A385583 (free).

Columns: A096267 (d=2), A365560 (d=3), A365562 (d=4), A365564 (d=5).

T(n,d) = Sum_{k=1..d} binomial(n,k)*A385582(n,k) (with A385582(n,k) = 0 if d > n).

A385583 Triangle read by rows: T(n,d) is the number of free d-dimensional polysticks of size n.

Original entry on oeis.org

1, 1, 2, 1, 5, 7, 1, 16, 28, 31, 1, 55, 160, 199, 205, 1, 222, 1085, 1651, 1768, 1779

Offset: 1

Pontus von Brömssen, Jul 04 2025

If d > n, there are T(n,n) such polysticks. The triangle only includes the values for d <= n.

			Triangle begins:
  n\d| 1   2    3    4    5    6
  ---+--------------------------
  1  | 1
  2  | 1   2
  3  | 1   5    7
  4  | 1  16   28   31
  5  | 1  55  160  199  205
  6  | 1 222 1085 1651 1768 1779

Index entries for sequences related to polyominoes.

Cf. A330891 (polyominoes), A365565 (main diagonal), A365566, A385581 (fixed).

Columns: A019988 (d=2), A365559 (d=3), A365561 (d=4), A365563 (d=5).

T(n,d) = Sum_{k=1..d} A365566(n,k).

cp's OEIS Frontend

User: Pontus von Brömssen

A387005 Number of free (d,2)-polyominoids of size n in arbitrary dimension d.

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A387002 Triangle read by rows: T(n,d) is the number of fixed, properly d-dimensional (d,2)-polyominoids of size n, 2 <= d <= n+1.

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A387004 Triangle read by rows: T(n,d) is the number of free, properly d-dimensional (d,2)-polyominoids of size n, 2 <= d <= n+1.

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A387003 Triangle read by rows: T(n,d) is the number of free (d,2)-polyominoids of size n, 2 <= d <= n+1.

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A386007 Least k such that there are exactly n primes that are popular on the interval [2,k] (see A385503); i.e., exactly n primes share the lead as the most common greatest prime factor of the numbers 2..k.

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Mathematica

A385654 Greatest prime factor of A385653(n).

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PARI

A385653 Least k such that A385652(k) = n.

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PARI

A385652 Maximum frequency of gpf(k) for 2 <= k <= n, where gpf(k) = A006530(k) is the greatest prime factor of k.

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A385581 Square array read by antidiagonals: T(n,d) is the number of fixed d-dimensional polysticks of size n.

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