Zach J. Shannon - cp's OEIS frontend

A354049 The smallest number that includes all the digits of n but does not equal n.

10, 10, 12, 13, 14, 15, 16, 17, 18, 19, 100, 101, 21, 31, 41, 51, 61, 71, 81, 91, 102, 12, 122, 32, 42, 52, 62, 72, 82, 92, 103, 13, 23, 133, 43, 53, 63, 73, 83, 93, 104, 14, 24, 34, 144, 54, 64, 74, 84, 94, 105, 15, 25, 35, 45, 155, 65, 75, 85, 95, 106, 16, 26, 36, 46, 56, 166, 76, 86, 96, 107

Offset: 0

Scott R. Shannon and Zach J. Shannon, May 16 2022

The terms cannot start with a leading zero so any number including a zero must have at least one digit greater than zero as its first digit. See the examples below.

			a(9) = 19 as there is no smaller number that includes the digit 9 but does not equal 9.
a(10) = 100 as there is no smaller number that includes the digits 1 and 0 but does not equal 10. Note that '01' = 1 is not allowed.
a(20) = 102 as there is no smaller number that includes the digits 2 and 0 but does not equal 20. Note that '02' = 2 is not allowed.
a(22) = 122 as there is no smaller number that includes two 2 digits but does not equal 22.
a(200) = 1002 as there is no smaller number that includes two 0 digits and the digit 2 but does not equal 200.

Scott R. Shannon, Table of n, a(n) for n = 0..9999
Scott R. Shannon, Image of the first 100000 terms. The green line is y = n.

Cf. A004185, A004185, A337227.

vd(n) = my(d=if (n, digits(n), [0])); vector(10, k, #select(x->(x==k-1), d));
isok(k, n, d) = if (k!=n, my(dd=vd(k)); for (i=1, #d, if (dd[i] < d[i], return(0))); return(1));
a(n) = my(k=0, d=vd(n)); while(!isok(k, n, d), k++); k; \\ Michel Marcus, May 17 2022

def ok(k, n):
    if k == n: return False
    sk, sn = str(k), str(n)
    return all(sk.count(d) >= sn.count(d) for d in set(sn))
def a(n):
    k = 0
    while not ok(k, n): k += 1
    return k
print([a(n) for n in range(71)]) # Michael S. Branicky, May 23 2022

A351784 Number of cells containing one or more grains of sand after n grains of sand are added to one cell in an initially empty and infinite 3D cubic grid for the 3D sandpile model.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 6, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 24, 25, 25, 25, 25, 25, 24, 25, 25, 25, 25, 25, 24, 25, 25, 25, 25, 25, 24, 25, 25, 25, 25, 25, 24, 25, 25, 25, 25, 25, 24, 25, 25, 25, 25, 25, 24, 25, 25, 25, 25, 25, 24, 25, 25

Offset: 0

Scott R. Shannon and Zach J. Shannon, Feb 19 2022

nonn

The 3D sandpile model follows the same rules as the 2D model except that cells topple and transfer one grain of sand to their six nearest neighbors when the cell contains 6 or more grains. Cells containing 0 to 5 grains are stable.

Scott R. Shannon, Table of n, a(n) for n = 0..10000
Per Bak, Chao Tang, and Kurt Wiesenfeld, Self-organized criticality: An explanation of the 1/f noise, Phys. Rev. Lett. 59 (1987), 381-384.
Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter, and Tianyuan Xu, Sandpiles and Dominos, Electronic Journal of Combinatorics, Volume 22(1), 2015.
Luis David Garcia-Puente and Brady Haran, Sandpiles, Numberphile video, YouTube.com, Jan. 13, 2017.
Zach J. Shannon, Image of the occupied cells for a(96)=44, bisected along the y-z plane. The colors are red=1 (24 total cells), blue=3 (14 total cells), violet=5 (6 total cells) grains per cell.

Cf. A351783, A351379, A349990 (2D version), A307652, A259013, A180230.

A351783 Number of grains of sand required to be added to one cell at the origin in an initially empty and infinite 3D cubic grid for the 3D sandpile model such that the distance from the origin of the furthest nonempty cell along the axes is n.

Original entry on oeis.org

0, 6, 36, 162, 516, 1230, 2430, 3756, 5862, 9036, 12822, 16710, 22182, 28758, 36144, 45444, 54966, 66270, 78870, 93834, 109866, 127260, 146412, 169698, 192366, 218214, 244752, 273480, 307224, 341430, 380988, 420558, 463350, 510024, 558090, 611088, 664494, 723060, 784014, 844134, 921486

Offset: 0

Scott R. Shannon and Zach J. Shannon, Feb 19 2022

nonn

The 3D sandpile model follows the same rules as the 2D model except that cells topple and transfer one grain of sand to their six nearest neighbors when the cell contains 6 or more grains. Cells containing 0 to 5 grains are stable.

Per Bak, Chao Tang, and Kurt Wiesenfeld, Self-organized criticality: An explanation of the 1/f noise, Phys. Rev. Lett. 59 (1987), 381-384.
Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter, and Tianyuan Xu, Sandpiles and Dominos, Electronic Journal of Combinatorics, Volume 22(1), 2015.
Luis David Garcia-Puente and Brady Haran, Sandpiles, Numberphile video, YouTube.com, Jan. 13, 2017.
Zach J. Shannon, Image of the occupied cells for a(40)=921486. For this and the below image, red=1, green=2, blue=3, violet=4, orange=5 grains per cell. The axes are in the middle of the red squares.
Zach J. Shannon, Image of the occupied cells for a(40)=921486, bisected along the y-z plane.

Cf. A351784, A351379, A307652, A259013, A180230.

A351379 The number of grains of sand in the identity element for the 3D sandpile group on an n X n X n cubic grid.

Original entry on oeis.org

24, 54, 288, 480, 744, 1062, 1968, 2616, 3480, 4398, 6000, 7344, 9744, 11628, 14256, 16632, 20376, 23436, 27312, 30984, 37104, 41652, 47424, 52776, 60432, 66636, 75552, 82752, 93288, 101676, 112488, 121968, 135768, 146436, 163032, 175182, 191256, 204690, 221784, 236646, 257400, 273738, 296784

Offset: 2

Scott R. Shannon and Zach J. Shannon, Feb 09 2022

nonn

The 3D sandpile model follows the same rules as the 2D model except that cells topple and transfer one grain of sand to their six nearest neighbors when the cell contains 6 or more grains. Cells containing 0 to 5 grains are stable.

See A307652 for details of the sandpile group identity.

			a(2) = 2 X 2 X 2 grid. Identity:
       Layer 1: | 3 3 |  Layer 2: | 3 3 |
                | 3 3 |           | 3 3 |  = 24 grains.
a(3) = 3 X 3 X 3 grid. Identity:
       Layer 1: | 3 2 3 |  Layer 2: | 2 1 2 |  Layer 3: | 3 2 3 |
                | 2 1 2 |           | 1 0 1 |           | 2 1 2 |
                | 3 2 3 |           | 2 1 2 |           | 3 2 3 |  = 54 grains.

Noah Doman, The Identity of the Abelian Sandpile Group, Bachelor Thesis, University of Groningen, January 2020.
Luis David Garcia-Puente and Brady Haran, Sandpiles, Numberphile video, YouTube.com, Jan. 13, 2017.
Yvan Le Borgne and Dominique Rossin, On the identity of the sandpile group, Discrete Mathematics, 256 (2002) 775-790.
Scott R. Shannon, Middle layer of the 100x100x100 identity. This contains 3486864 grains. For this and other images, white=0, red=1, green=2, blue=3, violet=4, yellow=5 grains per cell.
Scott R. Shannon, Top layer of the 100x100x100 identity.
Scott R. Shannon, Middle layer of the 101x101x101 identity. Similarly to the 2D sandpile model, when n is odd the middle layers have a cross-like pattern.
Zach J. Shannon, 3D image of the full 80x80x80 identity. The same colors as above are used except cells with no grains are shown as vacancies, not white.
Zach J. Shannon, 3D image of half the 80x80x80 identity.
Zach J. Shannon, 3D image of half the 80x80x80 identity without the cells containing 5 grains.
Zach J. Shannon, 3D image of half the 80x80x80 identity showing only the cells containing 5 grains.

Cf. A307652 (square grid), A259013, A180230, A300006, A007341.

Identity element = ([10n] - ([10n])*)* , where [10n] is the all 10's grid of size n X n X n, and (x)* represents the topple stabilization of the grid x.

The sequence is approximately fitted by the cubic a(n) ~ 3.48*n^3, where 3.48 corresponds to the approximate grains-per-cube density of the identity element configurations.

A348090 Place the numbers 1 to n on a square grid and sum both numbers in all created orthogonally adjacent pairs; a(n) gives the maximum possible value of the sum of all pair sums.

Original entry on oeis.org

0, 3, 9, 20, 34, 53, 75, 101, 134, 168, 204, 247, 293, 344, 399, 456, 518, 585, 654, 725, 803, 886, 978, 1065, 1154, 1252, 1355, 1467, 1572, 1679, 1797, 1920, 2052, 2188, 2315, 2444, 2586, 2733, 2889, 3049, 3198, 3349, 3515, 3686, 3866, 4050, 4238, 4413, 4590, 4784, 4983, 5191, 5403, 5619

Offset: 1

Scott R. Shannon and Zach J. Shannon, Oct 16 2021

nonn

On a square grid place the numbers 1 to n in any order or position. If any two numbers are orthogonally adjacent those two numbers are added, and the sum over all these pair sums is then found. The sequence gives the maximum possible value of this sum when placing numbers from 1 up to n.

Clearly, to maximize the sum all numbers should have one or more adjacent neighbors, and in general the larger numbers should be placed so that they appear in the most pairs so that their value contributes the most to the final sum. However, if two numbers have the same number of orthogonal neighbors then they can be switched since a number's contribution to the final sum is determined by the number of pairs it is in, not by the value of their pair neighbors.

			a(1) = 0 as the single number 1 has no neighbor to add to.
a(2) = 3 as the numbers 1 and 2 can be placed next to each other in one way, and the pair sum is 1+2 = 3.
a(3) = 9. The numbers 1,2,3 can be placed next to each other in six ways: 1-2-3, 1-3-2, 2-1-3, 2-3-1, 3-1-2, 3-2-1. The combinations with the largest pair sums are 1-3-2 and 2-3-1, the sum being (1+3)+(3+2) = 9. This is the largest sum as 3 is placed so that it is in two pairs and thus contributes twice to the sum.
a(4) = 20. The best way to arrange the numbers is in a 2 X 2 square. For example:
.
  1 2
  4 3
.
The sum is then (1+2)+(2+3)+(3+4)+(4+1) = 20. This is true for any permutation.
a(5) = 34. The best way to arrange the numbers is for 2,3,4,5 to be in a 2 X 2 square and for 1 to be placed next to 5. For example:
.
    2 3
  1 5 4
.
The sum is then (2+3)+(3+4)+(4+5)+(5+2)+(1+5) = 34.
a(6) = 53. The best way to arrange the numbers is in a 2 X 3 block where the 5 and 6 are in the middle of the long edge so that they both appear in three pairs. For example:
.
  2 6 4
  1 5 3
.
The sum is (2+6)+(6+4)+(1+5)+(5+3)+(2+1)+(6+5)+(4+3) = 53.

Cf. A346069 (multiplication), A003056, A005408.

A343208 a(n) = Sum_{k=1..n} kA001168(k)binomial(n-1,k-1), where A001168(k) is the number of fixed polyominoes with k cells.

Original entry on oeis.org

1, 5, 27, 143, 744, 3832, 19636, 100348, 511969, 2608905, 13282011, 67567527, 343510966, 1745495390, 8865633276, 45013599940, 228478238613, 1159398424925, 5881978415019, 29835289653043, 151308803657699, 767245632538063, 3889991549017581, 19720295705928713, 99961847384995974

Offset: 1

Scott R. Shannon and Zach J. Shannon, Apr 08 2021

nonn

This is the number of ways n blocks can be placed on a 2D grid such that, after the first block, each block touches at least one face of a previously placed block, and each block either touches the ground plane or is supported by a block below it. See the attached file for a derivation.

The number of ways n squares can be placed similarly on a 1D line is given by A001792.

			Considering the sequence as face-touching blocks:
a(1) = 1 as a single block can be placed in one way.
a(2) = 5 as, after the first block is placed, the second block can be placed so that it touches the ground plane and one of the four sides of the first block, or it can be placed directly on top of the first block, giving five total arrangements.
a(3) = 27 as the third block can be placed in one way directly on top of the tower of the two previous blocks, on the ground next to the tower of two blocks in four ways, next to one of the three faces of the second block on the ground plane or on top of the second block in 4*4 = 16 total ways, or on the ground plane touching one of the faces of the first block with the second block touching one of the other faces of the first block in 6 total ways. Summing the configurations gives 27 total ways the three blocks can be arranged.

Scott R. Shannon, Derivation of the title formula.

Cf. A001168, A001792, A007318.

A335066 Decimal numbers such that when they are written in all bases from 2 to 10 those numbers all share a common digit (the digit 0 or 1).

Original entry on oeis.org

1, 81, 91, 109, 127, 360, 361, 417, 504, 540, 541, 631, 661, 720, 781, 841, 918, 981, 991, 1008, 1009, 1039, 1080, 1081, 1088, 1089, 1090, 1091, 1093, 1099, 1105, 1111, 1116, 1117, 1118, 1119, 1120, 1121, 1122, 1123, 1124, 1125, 1126, 1128, 1134, 1135, 1136, 1137, 1138, 1139

Offset: 1

Scott R. Shannon and Zach J. Shannon, May 22 2020

As base 2 is included the only possible common digit between all the bases is either a 0 or 1.

			1 is a term as 1 written in all bases is 1.
81 is a term as 81_2 = 1010001, 81_3 = 10000, 81_4 = 1101, 81_5 = 311, 81_6 = 213, 81_7 = 144, 81_8 121, 81_9 = 100, 81_10 = 81, all of which contain the digit 1.
360 is a term as 360_2 = 101101000, 360_3 = 111100, 360_4 = 11220, 360_5 = 2420, 360_6 = 1400, 360_7 = 1023, 360_8 = 550, 360_9 = 550, 360_10 = 360, all of which contain the digit 0.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A335051, A258107, A145100, A317725, A181929, A331565, A153114.

def hasdigits01(n, b):
    has0, has1 = False, False
    while n >= b:
        n, r = divmod(n, b)
        if r == 0: has0 = True
        if r == 1: has1 = True
        if has0 and has1: return (True, True)
    return (has0, has1 or n==1)
def ok(n):
    all0, all1 = True, True
    for b in range(10, 1, -1):
        has0, has1 = hasdigits01(n, b)
        all0 &= has0; all1 &= has1
        if not all0 and not all1: return False
    return all0 or all1
print([k for k in range(1140) if ok(k)]) # Michael S. Branicky, May 23 2022

A335051 a(n) is the smallest decimal number > 1 such that when it is written in all bases from base 2 to base n those numbers all contain both 0 and 1.

Original entry on oeis.org

2, 9, 19, 28, 145, 384, 1128, 2601, 2601, 101256, 103824, 382010, 572101, 971400, 1773017, 1773017, 22873201, 64041048, 64041048, 1193875201, 2496140640, 10729882801, 21660922801, 120068616277, 333679563001, 427313653201, 427313653201, 10436523921264, 10868368953601

Offset: 2

Zach J. Shannon and Scott R. Shannon, May 21 2020

The sequence is infinite since 1 + lcm(2,...,n)^2 is always a candidate for a(n). - Giovanni Resta, May 24 2020

			a(3) = 9 as 9_2 = 1001 and 9_3 = 100, both of which contain a 0 and 1.
a(6) = 145 as 145_2 = 10010001, 145_3 = 12101, 145_4 = 2101, 145_5 = 1040, 145_6 = 401, all of which contain a 0 and 1.
a(9) = 2601 as 2601_2 = 101000101001, 2601_3 = 10120100, 2601_4 = 220221, 2601_5 = 40401, 2602_6 = 20013, 2601_7 = 10404, 2601_8 = 5051, 2601_9 = 3510, all of which contain a 0 and 1. Note that, as 2601 also contains a 0 and 1, a(10) = 2601.
a(16) = 1773017 as 1773017_2 = 110110000110111011001, 1773017_3 = 10100002010022, 1773017_4 = 12300313121, 1773017_5 = 423214032, 1773017_6 = 102000225, 1773017_7 = 21033101, 1773017_8 = 6606731, 1773017_9 = 3302108, 1773017_10 = 1773017, 1773017_11 = 1001104, 1773017_12 = 716075, 1773017_13 = 4A102C, 1773017_14 = 342201, 1773017_15 = 250512, 1773017_16 = 1B0DD9, all of which contain a 0 and 1.

Giovanni Resta, Table of n, a(n) for n = 2..32

Cf. A335066, A258107, A145100, A317725, A181929, A331565, A153114.

a[n_] := Block[{k=2}, While[ AnyTrue[ Range[n, 2, -1], ! SubsetQ[ IntegerDigits[k, #], {0, 1}] &], k++]; k]; a /@ Range[2, 13] (* Giovanni Resta, May 24 2020 *)

from numba import njit
@njit
def hasdigits01(n, b):
    has0, has1 = False, False
    while n >= b:
      n, r = divmod(n, b)
      if r == 0: has0 = True
      if r == 1: has1 = True
      if has0 and has1: return True
    return has0 and (has1 or n==1)
@njit
def a(n, start=2):
  k = start
  while True:
    for b in range(n, 1, -1):
      if not hasdigits01(k, b): break
    else: return k
    k += 1
anm1 = 2
for n in range(2, 21):
  an = a(n, start=anm1)
  print(an, end=", ")
  anm1 = an # Michael S. Branicky, Feb 09 2021

a(29)-a(30) from Giovanni Resta, May 24 2020

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User: Zach J. Shannon

A354049 The smallest number that includes all the digits of n but does not equal n.

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A351784 Number of cells containing one or more grains of sand after n grains of sand are added to one cell in an initially empty and infinite 3D cubic grid for the 3D sandpile model.

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A351783 Number of grains of sand required to be added to one cell at the origin in an initially empty and infinite 3D cubic grid for the 3D sandpile model such that the distance from the origin of the furthest nonempty cell along the axes is n.

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A351379 The number of grains of sand in the identity element for the 3D sandpile group on an n X n X n cubic grid.

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A348090 Place the numbers 1 to n on a square grid and sum both numbers in all created orthogonally adjacent pairs; a(n) gives the maximum possible value of the sum of all pair sums.

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A343208 a(n) = Sum_{k=1..n} k*A001168(k)*binomial(n-1,k-1), where A001168(k) is the number of fixed polyominoes with k cells.

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A335066 Decimal numbers such that when they are written in all bases from 2 to 10 those numbers all share a common digit (the digit 0 or 1).

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A335051 a(n) is the smallest decimal number > 1 such that when it is written in all bases from base 2 to base n those numbers all contain both 0 and 1.

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A343208 a(n) = Sum_{k=1..n} kA001168(k)binomial(n-1,k-1), where A001168(k) is the number of fixed polyominoes with k cells.