Danny Rorabaugh - cp's OEIS frontend

A322100 Number of minimal transitive permutation groups of degree n.

1, 1, 1, 2, 1, 4, 1, 5, 2, 6, 1, 17, 1, 6, 4, 75, 1, 23, 1, 47, 5, 6, 1, 213, 2, 7, 20, 30, 1, 79, 1, 12033, 3, 7, 4, 436, 1, 5, 4, 1963, 1, 84, 1, 148, 41, 4, 1

Offset: 1

Danny Rorabaugh, Nov 26 2018

A transitive group is minimal provided it has no proper transitive subgroups.

			There are two transitive groups of degree 3, A_3 and S_3, so A002106(3)=2. However, a(3)=1, because A_3 is minimal, but S_3 has proper transitive subgroups A_3 and S_2.

Derek Holt and Gordon Royle, A Census of Small Transitive Groups and Vertex-Transitive Graphs, arXiv:1811.09015 [math.CO], 2018.

Cf. A002106.

A294207 Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,k), 0 <= 3k <= 2n, that are below the line 3y=2x, only touching at the end points.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 3, 3, 1, 4, 7, 7, 1, 5, 12, 19, 19, 1, 6, 18, 37, 37, 1, 7, 25, 62, 99, 99, 1, 8, 33, 95, 194, 293, 293, 1, 9, 42, 137, 331, 624, 624, 1, 10, 52, 189, 520, 1144, 1768, 1768, 1, 11, 63, 252, 772, 1916, 3684, 5452, 5452

Offset: 0

Danny Rorabaugh, Oct 24 2017

			The table begins:
n=0: 1;
n=1: 1;
n=2: 1, 1;
n=3: 1, 2,  2;
n=4: 1, 3,  3;
n=5: 1, 4,  7,  7;
n=6: 1, 5, 12, 19,  19;
n=7: 1, 6, 18, 37,  37;
n=8: 1, 7, 25, 62,  99,  99;
n=9: 1, 8, 33, 95, 194, 293, 293.

Eric Weisstein's World of Mathematics, Lattice Path.

Cf. A009766, A293946.

T[, 0] = 1; T[n, k_] := T[n, k] = Which[0 < k < 2(n-1)/3, T[n-1, k] + T[n, k-1], 2(n-1) <= 3k <= 2n, T[n, k-1]];
Table[T[n, k], {n, 0, 15}, {k, 0, Floor[2n/3]}] // Flatten (* Jean-François Alcover, Jul 10 2018 *)

T = [[1]]
for n in range(1,15):
    T.append([T[-1][0]])
    for k in range(1,floor(2*n/3) + 1):
        T[-1].append(T[-1][k-1])
        if 2*(n-1)>3*k:
            T[-1][-1] += T[-2][k]

T(n,0) = 1; for 0 < k < 2(n-1)/3, T(n,k) = T(n-1,k) + T(n,k-1); for 2(n-1) <= 3k <= 2n, T(n,k) = T(n,k-1).

A270952 T(n, k) is the number of k-element connected subposets of the n-th Boolean lattice, 0 <= k <= 2^n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 5, 4, 1, 1, 8, 19, 42, 61, 56, 28, 8, 1, 1, 16, 65, 304, 1129, 3200, 6775, 10680, 12600, 11386, 8002, 4368, 1820, 560, 120, 16, 1, 1, 32, 211, 1890, 14935, 97470

Offset: 0

Danny Rorabaugh, Mar 26 2016

The n-th Boolean lattice is the set of all subsets of {1,2,...,n}, partially ordered by inclusion.

			The triangle begins:
n\k 0  1   2    3     4     5    6     7     8     9   10   11   12  13  14
0   1  1
1   1  2   1
2   1  4   5    4     1
3   1  8  19   42    61    56   28     8     1
4   1 16  65  304  1129  3200 6775 10680 12600 11386 8002 4368 1820 560 120
5   1 32 211 1890 14935 97470  ...
For T(2, 2) = 5: [{},{1}], [{},{2}], [{},{1,2}], [{1},{1,2}], [{2},{1,2}].

Eric Weisstein's World of Mathematics, Boolean Algebra.

Columns: A000012 (k = 0, 2^n), A000079 (k = 1, 2^n - 1), A001047 (k = 3).

def ConnectedSubs(n): # Returns row n of T(n, k).
    Bn = posets.BooleanLattice(n)
    counts = [0]*(2^n+1)
    for X in Subsets(range(2^n)):
        if Bn.subposet(X).is_connected():
            counts[len(X)] += 1
    return counts

A270440 Least k such that binomial(k, 2) >= binomial(2*n, n).

Original entry on oeis.org

2, 3, 4, 7, 13, 23, 44, 84, 161, 313, 609, 1189, 2327, 4562, 8958, 17614, 34673, 68318, 134724, 265878, 525066, 1037554, 2051390, 4057939, 8030892, 15900354, 31493446, 62400953, 123682583, 245223436, 486342641, 964809156, 1914483817, 3799849586, 7543612064, 14979070587, 29749371096, 59095356237, 117410567231

Offset: 0

Danny Rorabaugh, Mar 17 2016

nonn

Open question: Does binomial(a(n), 2) = binomial(2*n, n) for any n > 2? An affirmative answer would settle whether there exists an odd term greater than 3 in A003016.

binomial(a(n),2) > binomial(2*n,n) for 2 < n <= 800000. - Chai Wah Wu, Mar 22 2016

Chai Wah Wu, Table of n, a(n) for n = 0..1000

Cf. A000217, A000984, A003015, A100967.

Table[SelectFirst[Range[10^7], Binomial[#, 2] >= Binomial[2 n, n] &], {n, 0, 22}] (* Michael De Vlieger, Mar 17 2016, Version 10 *)

a(n) = {my(c = binomial(2*n, n)); my(k = 0); while (binomial(k,2) < c, k++); k;} \\ Michel Marcus, Mar 17 2016

from _future_ import division
from gmpy2 import iroot
A270440_list, b = [], 8
for n in range(1001):
    q, r = iroot(b+1,2)
    A270440_list.append(int((q+1)//2 + (0 if r else 1)))
    b = b*2*(2*n+1)//(n+1) # Chai Wah Wu, Mar 22 2016

def k2_2nn(M): # Produces the first M terms.
    K, n, center, k, triangle = [], 0, 1, 1, 0
    while len(K)

Conjecture: a(n) ~ 2^(n + 1/2) / (Pi*n)^(1/4). - Vaclav Kotesovec, Mar 23 2016

a(n) = ceiling(((8*binomial(2*n,n)+1)^(1/2)+1)/2). The above conjecture is true asymptotically. Using Stirling's formula for the approximation of n!, we get binomial(2*n,n) ~ 2^(2*n)/(Pi*n)^(1/2) and inserting this in the formula for a(n) results in the above approximation for a(n). - Chai Wah Wu, Mar 23 2016

A269832 Greatest term of height n in Recamán's sequence A005132.

Original entry on oeis.org

1, 3, 7, 13, 25, 46, 91, 164, 286, 515, 962, 1744, 3137, 5810, 10319, 18953, 35079, 63237

Offset: 1

Danny Rorabaugh, Mar 05 2016

The height (A064289) of a term in Recamán's sequence (A005132) = number of addition steps - number of subtraction steps to produce it.

Index entries for sequences related to Recamán's sequence

Cf. A064289, A064290, A064293, A269830, A269831.

A269831 Least term of height n in Recamán's sequence A005132.

Original entry on oeis.org

1, 2, 6, 8, 14, 26, 4, 47, 92, 111, 181, 150, 371, 361, 781, 828, 366, 19

Offset: 1

Danny Rorabaugh, Mar 05 2016

The height (A064289) of a term in Recamán's sequence (A005132) = number of addition steps - number of subtraction steps to produce it.

Index entries for sequences related to Recamán's sequence

Cf. A064289, A064290, A064293, A269830, A269832.

A269830 Number of terms of height n in Recamán's sequence A005132.

Original entry on oeis.org

1, 2, 2, 6, 11, 22, 34, 61, 115, 220, 397, 681, 1329, 2430, 4561, 8116, 14848, 24878

Offset: 1

Danny Rorabaugh, Mar 05 2016

The height (A064289) of a term in Recamán's sequence (A005132) = number of addition steps - number of subtraction steps to produce it.

Danny Rorabaugh, All terms of height n<=18 in Recamán's sequence
Index entries for sequences related to Recamán's sequence

Cf. A064289, A064290, A064293.

A269699 Irregular triangle read by rows: T(n, k) is the number of k-element proper ideals of the n-dimensional Boolean lattice, with 0 < k < 2^n.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 3, 4, 3, 3, 1, 1, 4, 6, 10, 13, 18, 19, 24, 19, 18, 13, 10, 6, 4, 1, 1, 5, 10, 20, 35, 61, 95, 155, 215, 310, 387, 470, 530, 580, 605, 621, 605, 580, 530, 470, 387, 310, 215, 155, 95, 61, 35, 20, 10, 5, 1, 1, 6, 15, 35, 75, 156, 306, 605, 1110, 2045, 3512, 5913, 9415

Offset: 1

Danny Rorabaugh, Mar 03 2016

The set of maximal elements of an ideal is an antichain; conversely, the down-set of a nonempty antichain is an ideal. The down-set of the top element of the n-dimensional Boolean lattice contains all 2^n elements of the lattice, and thus is not a proper ideal.
Empirically, the rows are unimodal.
By the Markowsky paper, T(n, k) = T(n, 2^n - k).
Also, T(n,k) is the number of n-dimensional Ferrers diagrams with k nodes (i.e., (n-1)-dimensional partitions) that fit into an n-dimensional hypercube of side 2 (i.e., a Boolean or binary hupercube). T(n, k) = T(n, 2^n - k) follows from the map that takes a Ferrers diagram to its complement in the box. - Suresh Govindarajan, Apr 10 2016

			For row n = 3, the k-element proper ideals are the down-sets of the following antichains:
T(3, 1) = 1: [{}];
T(3, 2) = 3: [{0}], [{1}], [{2}];
T(3, 3) = 3: [{0},{1}], [{0},{2}], [{1},{2}];
T(3, 4) = 4: [{0,1}], [{0,2}], [{1,2}], [{0},{1},{2}];
T(3, 5) = 3: [{0,1},{2}], [{0,2},{1}], [{1,2},{0}];
T(3, 6) = 3: [{0,1},{0,2}], [{0,1},{1,2}], [{0,2},{1,2}];
T(3, 7) = 1: [{0,1},{0,2},{1,2}].
E.g., the 5-element down-set of [{0,1},{2}] is [{},{0},{1},{2},{0,1}].
The table begins:
n\k 1 2  3  4  5  6  7   8   9  10  11  12  13  14  15  16  17
1   1
2   1 2  1
3   1 3  3  4  3  3  1
4   1 4  6 10 13 18 19  24  19  18  13  10   6   4   1
5   1 5 10 20 35 61 95 155 215 310 387 470 530 580 605 621 605 ...

Danny Rorabaugh and Suresh Govindarajan, Table of n, a(n) for n = 1..279
George Markowsky, The level polynomials of the free distributive lattices, Discrete Math., 29 (1980), 275-285.
Wikipedia, Boolean algebra and Ideal.

Columns are: A000012 (k = 1), A000027 (k = 2), A000217 (k = 3), A000292 (k = 4), A095661 (k = 5).

Cf. A007153 (row sums), A007318, A059119.

# Returns row n.
def T(n):
  B = posets.BooleanLattice(n)
  t = [0]*(2^n + 1)
  for A in B.antichains():
    t[len(B.order_ideal(A))] += 1
  return t[1:-1]

A261147 English flagpole sequence of nonnegative integers (like A215693, but with a(1)=0).

Original entry on oeis.org

0, 1, 3, 2, 4, 7, 5, 8, 13, 14, 9, 10, 12, 20, 21, 15, 22, 24, 23, 6, 11, 17, 16, 18, 25, 19, 27, 26, 28, 29, 38, 30, 31, 32, 33, 34, 35, 36, 37, 39, 41, 40, 42, 44, 43, 45, 47, 48, 49, 51, 46, 57, 53, 50, 55, 59, 52, 62, 58, 111, 65, 61, 54, 72, 63, 67, 56

Offset: 1

Danny Rorabaugh, Nov 28 2015

Concatenate the spelling of the terms in English (ignoring "and", punctuation, capitalization, and spaces); the i-th letter of this infinite word is in a(i). This is the lexicographically least such sequence of distinct nonnegative integers. A215693 is the lexicographically least such sequence of distinct positive integers.

In the first 100000 terms: the records for a(n)-n are set by a(1)=0, a(3)=3, a(6)=7, a(9)=13, a(14)=20, a(31)=38, a(60)=111, a(119)=211, a(18617)=18711; the records for n-a(n) are set by a(1)=0, a(4)=2, a(20)=6, a(335)=319.

			The sequence begins: 0,1,3,2,4,7,5,8,13....
The initial terms are spelled: ZERO;ONE;THREE;TWO;....
These letters are found in the spelling of the initial terms: Zero, onE, thRee, twO; fOur, seveN, fivE; eighT, tHirteen, ....

Danny Rorabaugh, Table of n, a(n) for n = 1..3000
Wikipedia, English numerals
Hans Havermann, List of a(n) for n = 1..100000

Cf. A215693.

little = {0:"", 1:"one", 2:"two", 3:"three", 4:"four", 5:"five", 6:"six", 7:"seven", 8:"eight", 9:"nine", 10:"ten", 11:"eleven", 12:"twelve", 13:"thirteen", 14:"fourteen", 15:"fifteen", 16:"sixteen", 17:"seventeen", 18:"eighteen", 19:"nineteen"}
decade = {2:"twenty", 3:"thirty", 4:"forty", 5:"fifty", 6:"sixty", 7:"seventy", 8:"eighty", 9:"ninety"}
illion = {1:"thousand", 2:"million", 3:"billion", 4:"trillion"}
def nmb_wrd(n): # For positive integers n<10^15
    if n<20: return little[n]
    if n<100: return decade[floor(n/10)] + little[n%10]
    if n<1000: return little[floor(n/100)] + "hundred" + nmb_wrd(n%100)
    k = floor((len(str(n))-1)/3)
    return nmb_wrd(floor(n/10^(3*k))) + illion[k] + nmb_wrd(n%(10^(3*k)))
def English_fp(n, A = [0,1,3,2], i = 1):
    while len(A)A215693 is English_fp(113, [1,7,3])

Corrected by Danny Rorabaugh, Nov 29 2015

A264815 Semirps: a semirp (or semi-r-p) is a semiprime r*p with r and p both reversed primes.

Original entry on oeis.org

4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 39, 49, 51, 55, 62, 65, 74, 77, 85, 91, 93, 111, 119, 121, 142, 143, 146, 155, 158, 169, 185, 187, 194, 202, 213, 214, 217, 219, 221, 226, 237, 259, 262, 289, 291, 298, 302, 303, 314, 321, 334, 339, 341, 355

Offset: 1

Danny Rorabaugh, Nov 25 2015

A semiprime (A001358) is the product of two prime, not necessarily distinct. A semiprime is in this list if those two primes (A000040) are reversed primes (A004087).

Since A007500 is the intersection of A000040 and A004087, this sequence is also the sorted list of all r*p with r and p in A007500.

			9 is in the list because 9 = 3*3 is a semiprime and reverse(3) = 3 is prime.
143 is in the list because 143 = 11*13 is a semiprime and both reverse(11) = 11 and reverse(13) = 31 are prime.

Danny Rorabaugh, Table of n, a(n) for n = 1..10000

Cf. A001358, A006567, A007500, A097393, A109019, A115670.

With[{nn=250},Take[Union[Times@@@Select[Tuples[IntegerReverse/@Prime[Range[nn]],2],AllTrue[#,PrimeQ]&]],60]] (* Harvey P. Dale, Apr 27 2025 *)

reverse = lambda n: sum([10^i*int(str(n)[i]) for i in range(len(str(n)))])
def is_semirp(n):
  F = factor(n)
  if sum([f[1] for f in F])==2:
    r, p = F[0][0], F[-1][0]
    if is_prime(reverse(r)) and is_prime(reverse(p)): return True
[a for a in range(1,356) if is_semirp(a)] # Danny Rorabaugh, Nov 25 2015

[A007500]^2, sorted.

cp's OEIS Frontend

User: Danny Rorabaugh

A322100 Number of minimal transitive permutation groups of degree n.

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A294207 Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,k), 0 <= 3k <= 2n, that are below the line 3y=2x, only touching at the end points.

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A270952 T(n, k) is the number of k-element connected subposets of the n-th Boolean lattice, 0 <= k <= 2^n.

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A270440 Least k such that binomial(k, 2) >= binomial(2*n, n).

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A269832 Greatest term of height n in Recamán's sequence A005132.

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A269831 Least term of height n in Recamán's sequence A005132.

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A269830 Number of terms of height n in Recamán's sequence A005132.

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A269699 Irregular triangle read by rows: T(n, k) is the number of k-element proper ideals of the n-dimensional Boolean lattice, with 0 < k < 2^n.

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A261147 English flagpole sequence of nonnegative integers (like A215693, but with a(1)=0).

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