A008615 -id:A008615 - cp's OEIS frontend

A274342 Irregular triangle read by rows, giving the numerators of the coefficients of the Eisenstein series G_{2n} multiplied by 2n-1, for n >= 2. Also Laurent coefficients of Weierstrass's P function.

1, 1, 1, 3, 1, 2, 2, 60, 5, 1, 29, 485, 2, 1722, 5446, 3, 8000, 10, 5300, 270, 181188, 955290, 4, 4, 15988040, 416012, 32420068, 2682744, 223, 25851, 8409205, 49871, 301, 1713301109422, 1066033105795, 4270, 57425882, 859704866, 11125766, 77746116, 39343318862281, 501010332520, 4762

Offset: 2

Wolfdieter Lang, Jun 20 2016

The length of row n is A008615(n), n >= 2.
The denominator triangle is given in A274343.
The Eisenstein series with even index, G_{2*n}, when multiplied by 2*n-1, namely c(n) := (2*n-1)*G_{2*n}, satisfy the well-known recurrence relation (n-3) * (2*n +1) * c(n) = 3 * Sum_{j=2..n-2} c(j) * c(n-j), for  n >= 4, with initial terms c(2) = c2 and c(3) = c3. See, e.g., the references Abramowitz-Stegun, 18.5.3, p. 635, Apostol  p. 13, and  Tricomi, p. 34.
The solution of this recurrence is c(n) = Sum a(n, m)/A274343(n, m)*c2^e2(n, m)*c3^e3(n, m), where the sum is over the partitions of n with parts 2 and 3 only, and with nonnegative exponents e2(n, m) and e3(n, m), where m = 1..A008615(n). The order is by increasing number of parts. E.g., n=6 with the partitions 3^2 and 2^3, with c(6) = (1/13)*c(3)^2 + (2/39)*c(2)^3. See also the Abramowitz-Stegun reference 18.5.9 - 18.5.24, p. 636, for n=4..19, but not given in lowest terms, and with decreasing number of parts for the partitions (contrary to the listing of partitions on p. 831).
The rational numbers c(n) appear also as coefficients in the Laurent series of Weierstrass's P function: WeierstrassP(z; g_2, g_3) = 1/z^2 + Sum_{n >= 2} c(n) * z^{2*n-2}, with g_2 = 20*c(2) and g_4 = 28*c(3). See, e.g., the Abramowitz-Stegun reference 18.5.1, p. 635. See also the o.g.f. given below.
For the polynomials c(2)..c(20) see the W. Lang link, also for the corresponding Eisenstein series G_{2*n} in terms of g_2 and g_4.

			The irregular triangle a(n, m) begins:
n\m          1          2         3   ...
2:           1
3:           1
4:           1
5:           3
6:           1          2
7:           2
8:          60          5
9:           1         29
10:        485          2
11:       1722       5446
12:          3       8000        10
13:       5300        270
14:     181188     955290         4
15:          4   15988040    416012
16:   32420068    2682744       223
17:      25851    8409205     49871
...
row n = 18: 301  1713301109422 1066033105795 4270,
row n = 19: 57425882 859704866 11125766,
row n = 20: 77746116 39343318862281 501010332520  4762.
The irregular triangle of rationals r(n) starts:
n\m:      1              2            3  ...
2:       1/1
3:       1/1
4:       1/3
5:       3/11
6:       1/13           2/39
7:       2/33
8:      60/2431         5/663
9:       1/2           29/2717
10:    485/80223        2/1989
11:   1722/1062347   5446/3187041
12:     3/16055      8000/6605027   10/77571
13:  5300/11685817   270/1062347
...
row n = 14: 181188/2002524095 955290/4405553009  4/249951,
row n = 15: 4/497705  15988040/155409680283 416012/11559397707,
row n = 16: 32420068/1123416017295 2682744/74894401153  223/114727509,
row n = 17:  25851/5643476995    8409205/409716429837 49871/10158258591,
row n = 18: 301/909705199  1713301109422/233400836858808047  1066033105795/190964321066297493  4270/18394643943,
row n = 19: 57425882/34825896536145  859704866/229850917138557  11125766/17096349208653,
row n = 20: 77746116/357856262339147  39343318862281/24291640943843637507  501010332520/602272089516784401  4762/174041631153.

T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990, p. 13.
F. Tricomi, Elliptische Funktionen (German translation by M. Krafft of: Funzioni ellittiche), Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1948, pp. 34-35.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], ch. 18.5, pp. 635-636.
Wolfdieter Lang, Rationals c(n), n = 2..20, and Eisenstein series G_{2*k}, k = 2..10.

Cf. A008615, A274343.

a(n) = numerator(r(n)) with the rationals r(n) in lowest terms obtained from the c(n) recurrence given in a comment above as coefficients of powers of c2 and c3 corresponding to the partitions of n with parts 2 and 3 only, when sorted with increasing number of parts.
O.g.f: C(x) = Sum_{n >= 2} c(n)*x^n = x*WeierstrassP(sqrt(x), g_2 = 20*c(2), g_3 = 28*c(3)) - 1. Compare with Abramowitz-Stegun, 18.5.1, p. 635.
Nonlinear differential equation of second order for the o.g.f C(x) derived from the recurrence relation of c(n): 2*x^2*(d^2/dx^2)C(x) - 3*x*(d/dx)C(x) - 3*C(x) + 5*x^2*c(2) - 3*C(x)^2 = 0, with C(0) = 0 and C'(0) = 0.

A340010 The order of the largest weakly connected component of the Collatz digraph of order n.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 3, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 12, 12, 12, 12, 13, 13, 21, 21, 22, 22, 22, 22, 24, 24, 25, 25, 26, 26, 26, 26, 27, 27, 28, 28, 33, 33, 33, 33, 34, 34, 36, 36, 37, 37, 37, 37, 39, 39, 40, 40, 40, 40, 40, 40, 41, 41, 42, 42, 44, 44

Offset: 1

Sebastian Karlsson, Dec 26 2020

nonn

The Collatz digraph of order n is the directed graph with the vertex set V = {1, 2, ..., n} and the arrow set A = {m -> A014682(m) | m and A014682(m) are elements of V}.

			The weakly connected components of the Collatz digraph of order 3 are 1 -> 2 -> 1 and the singleton 3. The order of the largest component is #{1, 2} = 2.
The weakly connected components of the Collatz digraph of order 10 correspond to the following partition of {1, 2, ..., 10}: {1, 2, 3, 4, 5, 6, 8, 10}, {7} and {9}. The order of the largest component is #{1, 2, 3, 4, 5, 6, 8, 10} = 8. Hence, a(10) = 8.
The weakly connected components of the Collatz digraph of order 20 correspond to the partition {1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 16, 20}, {7, 9, 11, 14, 17, 18}, {15} and {19}. The order of the largest component is #{1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 16, 20} = 12. Thus, a(20) = 12.

J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010.

Sebastian Karlsson, Table of n, a(n) for n = 1..20000
Lorenzo Sauras Altuzarra, Some arithmetical problems that are obtained by analyzing proofs and infinite graphs, arXiv:2002.03075 [math.NT], 2020.
Thijs Laarhoven, The 3n+1 conjecture, Eindhoven University of Technology, Bachelor thesis (2009). See also.
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006370, A014682, A127824, A248573, A088975, A008615, A103469.

Cf. A340985.

import networkx as nx
def T(n): #A014682
    return n//2 if n%2 == 0 else (3*n+1)//2
def a(n):
    G = nx.Graph()
    G.add_nodes_from(range(1, n+1))
    G.add_edges_from([(m,T(m)) for m in range(1, n+1) if T(m) <= n])
    return len(max(nx.connected_components(G)))
for n in range(1, 70):
    print(a(n), end=", ")

A010764 a(n) = floor(n/2) mod floor(n/3).

Original entry on oeis.org

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

Offset: 3

N. J. A. Sloane, Simon Plouffe

			G.f. = x^6 + x^7 + x^9 + 2*x^10 + 2*x^11 + 2*x^12 + 2*x^13 + 3*x^14 + ...

Antti Karttunen, Table of n, a(n) for n = 3..10000

Cf. A002264, A004526, A008615.

[ seq(floor(n/2) mod floor(n/3), n=3..100) ];

Table[Mod[Floor[n/2],Floor[n/3]],{n,3,100}] (* Harvey P. Dale, Apr 06 2018 *)

{a(n) = if( n<3, 0, (n\2) % (n\3))}; /* Michael Somos, Feb 06 2003 */

G.f.: x^6*(1 + x - x^2 - x^3 + x^4 + 2*x^5 - 2*x^7)/((1 - x^2)*(1 - x^3)).
a(n) = A008615(n) if n>8. - Michael Somos, Feb 06 2003
a(n) = A004526(n) mod A002264(n). - Antti Karttunen, Aug 23 2017

A026806 a(n) = number of numbers k such that only one partition of n has least part k.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1).

Cf. A008615.

List([1..90], n-> 1+Int(n/2)-Int(n/3) ); # G. C. Greubel, Nov 09 2019

[1+Floor(n/2)-Floor(n/3): n in [1..90]]; // G. C. Greubel, Nov 09 2019

seq(1+floor(n/2)-floor(n/3), n = 0..90); # G. C. Greubel, Nov 09 2019

Rest[CoefficientList[Series[x(1+2x-x^3-x^4)/((1-x^2)(1-x^3)), {x,0,90}], x]]  (* Harvey P. Dale, Apr 22 2011 *)
Table[1 + Floor[n/2] - Floor[n/3], {n, 90}] (* G. C. Greubel, Nov 09 2019 *)

PARI
```
a(n)=if(n<1,0,1+(n\2)-(n\3))
```

[1+floor(n/2)-floor(n/3) for n in (1..40)] # G. C. Greubel, Nov 09 2019

G.f.: x*(1+2*x-x^3-x^4)/((1-x^2)*(1-x^3)).

a(n) = A008615(n+6) = 1 + A008615(n), n>0.

A060549 a(n) is the number of distinct patterns (modulo geometric D3-operations) with strict median-reflective (palindrome) symmetry (i.e., having no other symmetry) which can be formed by an equilateral triangular arrangement of closely packed black and white cells satisfying the local matching rule of Pascal's triangle modulo 2, where n is the number of cells in each edge of the arrangement. The matching rule is such that any elementary top-down triangle of three neighboring cells in the arrangement contains either one or three white cells.

Original entry on oeis.org

0, 1, 2, 2, 6, 6, 12, 14, 28, 28, 60, 60, 120, 124, 248, 248, 504, 504, 1008, 1016, 2032, 2032, 4080, 4080, 8160, 8176, 16352, 16352, 32736, 32736, 65472, 65504, 131008, 131008, 262080, 262080, 524160, 524224, 1048448, 1048448, 2097024, 2097024

Offset: 1

André Barbé (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 03 2001

Harry J. Smith, Table of n, a(n) for n=1..500
A. Barbé, Symmetric patterns in the cellular automaton that generates Pascal's triangle modulo 2, Discr. Appl. Math. 105(2000), 1-38.
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (0,2,0,0,0,2,0,-4).

Cf. A060546, A060548, A008619, A008615.

a(n) = { 2^ceil(n/2) - 2^(floor((n + 3)/6) + (n%6==1)) } \\ Harry J. Smith, Jul 07 2009

a(n) = 2^ceiling(n/2) - 2^(floor((n+3)/6) + d(n)), with d(n)=1 if n mod 6=1 else d(n)=0.
a(n) = A060546(n) - A060548(n) = 2^A008619(n-1) - 2^A008615(n+1), for n >= 1.
G.f.: x^2*(2*x^4 + 2*x^3 + 2*x + 1) / ((2*x^2-1)*(2*x^6-1)). - Colin Barker, Aug 29 2013

A060551 a(n) is the number of nonsymmetric patterns (no reflective, nor rotational symmetry) which may be formed by an equilateral triangular arrangement of closely packed black and white cells satisfying the local matching rule of Pascal's triangle modulo 2, where n is the number of cells in each edge of the arrangement. The matching rule is such that any elementary top-down triangle of three neighboring cells in the arrangement contains either one or three white cells.

Original entry on oeis.org

0, 0, 0, 6, 12, 42, 84, 210, 420, 924, 1860, 3900, 7800, 15996, 31992, 64728, 129528, 260568, 521136, 1045464, 2090928, 4187952, 8376240, 16764720, 33529440, 67084080, 134168160, 268385376, 536772192, 1073642592, 2147285184, 4294769760, 8589539520, 17179472064

Offset: 1

André Barbé (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 03 2001

Harry J. Smith, Table of n, a(n) for n=1..500
A. Barbé, Symmetric patterns in the cellular automaton that generates Pascal's triangle modulo 2, Discr. Appl. Math. 105(2000), 1-38.
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (2,2,-2,-4,-4,10,-4,-4,4,8,8,-16).

Cf. A000079, A060546, A060547, A060548, A008619, A008611, A008615.

LinearRecurrence[{2,2,-2,-4,-4,10,-4,-4,4,8,8,-16},{0,0,0,6,12,42,84,210,420,924,1860,3900},40] (* Harvey P. Dale, Feb 01 2015 *)

a(n) = { 2^n-3*2^ceil(n/2)-2^(floor(n/3)+(n%3)%2)+3*2^(floor((n+3)/6)+(n%6==1)) } \\ Harry J. Smith, Jul 07 2009

a(n) = 2^n - 3*2^ceiling(n/2) - 2^(floor(n/3)+(n mod 3)mod 2) + 3*2^(floor((n+3)/6) + d(n)), with d(n)=1 if n mod 6=1 else d(n)=0.
a(n) = A000079(n) - 3*A060546(n) - A060547(n) + 3*A060548(n).
a(n) = A000079(n) - 3*2^A008619(n-1) - 2^A008611(n-1) + 3*2^A008615(n+1), for n >= 1.
G.f.: -6*x^4*(2*x^6 + 2*x^5 - x^4 + 2*x^3 - x^2 - 1) / ((2*x-1)*(2*x^2-1)*(2*x^3-1)*(2*x^6-1)). - Colin Barker, Aug 29 2013
a(n) = 6*A060552(n). - Andrew Howroyd, Dec 24 2024

More terms from Colin Barker, Aug 29 2013

A274343 Irregular triangle read by rows giving the denominators of the coefficients of the Eisenstein series G_{2n} multiplied by 2n-1, for n >= 2. Also Laurent coefficients of Weierstrass's P function.

Original entry on oeis.org

1, 1, 3, 11, 13, 39, 33, 2431, 663, 247, 2717, 80223, 1989, 1062347, 3187041, 16055, 6605027, 77571, 11685817, 1062347, 2002524095, 4405553009, 247, 2717, 497705, 155409680283, 11559397707, 1123416017295, 74894401153, 114727509, 5643476995, 409716429837, 10158258591, 909705199, 233400836858808047, 190964321066297493, 18394643943, 34825896536145, 229850917138557, 17096349208653, 357856262339147, 24291640943843637507, 602272089516784401, 174041631153

Offset: 2

Wolfdieter Lang, Jun 20 2016

The length of row n is A008615(n), n >= 2.
The numerator triangle is given in A274342 where also details and references are given.
a(n) = denominator(r(n)) where the rationals r(n) are reduced to lowest terms obtained from the c(n) recurrence given in a comment of A274342 as coefficients of powers of c2 and c3 corresponding to the partitions of n with parts 2 and 3 only, when sorted with increasing number of parts.

			The irregular triangle a(n, m) begins:
n\m             1            2            3
2:              1
3:              1
4:              3
5:             11
6:             13           39
7:             33
8:           2431          663
9:            247         2717
10:          8022         1989
11:       1062347      3187041
12:         16055      6605027        77571
13:      11685817      1062347
14:    2002524095   4405553009       249951
15:        497705 155409680283  11559397707
16: 1123416017295  74894401153    114727509
17:    5643476995 409716429837  10158258591
...
row n = 18: 909705199 233400836858808047 190964321066297493 18394643943,
row n = 19: 34825896536145  229850917138557 17096349208653,
row n = 20: 357856262339147 24291640943843637507 602272089516784401 174041631153.
...
For the rationals r(n), n = 2..20, see A274342.

Cf. A274342.

A300185 Irregular triangle read by rows: T(n, {j,k}) is the number of partitions of n that have exactly j parts equal to k; 1 <= j <= n, 1 <= k <= n.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 1, 1, 0, 1, 2, 1, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 1, 1, 0, 1, 3, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 4, 2, 2, 1, 1, 0, 1, 4, 2, 1, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 3, 0

Offset: 1

J. Stauduhar, Feb 27 2018

Row sums = A027293.
If superfluous zeros are removed from the right side of each row, the row lengths = 1,2,1,3,1,1,4,2,... = A010766.
Sum of each N X N block of rows = 1,2,4,7,12,19,... = A000070.
The sum of the partitions of n that are over-counted in each block of N x N rows = A000070(n) - A000041(n) = A058884(n), n >= 1.
Concatenation of first row from each N X N block = A116598.
As noted by Joerg Arndt in A116598, the first row from each N X N block in reverse converges to A002865.  Two sequences emerge from alternating second rows in reverse:  for 2n, converges to even-indexed terms in A027336, and for 2n+1, converges to odd-indexed terms in A027336.
Counting the rows in each N X N block where columns j=2 > 0 and j=3 through j=n are all zeros produces A008615(n), n > 0.

			      \ j  1 2 3 4 5
     k
n
1:   1     1
2:   1     0 1
     2     1 0
3:   1     1 0 1
     2     1 0 0
     3     1 0 0
4:   1     1 1 0 1
     2     1 1 0 0
     3     1 0 0 0
     4     1 0 0 0
5:   1     2 1 1 0 1
     2     2 1 0 0 0
     3     2 0 0 0 0
     4     1 0 0 0 0
     5     1 0 0 0 0
.
.
.

J. Stauduhar, Table of n, a(n) for n = 1..10000
Jerome Kelleher and Barry O'Sullivan, Generating All Partitions: A Comparison Of Two Encodings, arXiv:0909.2331 [cs.DS], 2009-2014.
J. Stauduhar, Original Python program.

Cf. A000041, A000070, A008615, A010766, A027293, A027336, A058884, A116598.

Array[With[{s = IntegerPartitions[#]}, Table[Count[Map[Count[#, k] &, s], j], {k, #}, {j, #}]] &, 7] // Flatten (* Michael De Vlieger, Feb 28 2018 *)

Python
```
# See Stauduhar link.
```

A366154 Irregular triangle read by rows: T(n,k) is the number of integer partitions of n with at least one part a_i such that a_i - a_{i+k} = k.

Original entry on oeis.org

0, 1, 2, 3, 1, 5, 1, 7, 3, 1, 11, 3, 2, 15, 7, 3, 1, 22, 9, 4, 2, 30, 15, 7, 4, 1, 42, 20, 11, 6, 2, 56, 32, 16, 9, 4, 1, 77, 40, 22, 12, 7, 2, 101, 61, 33, 19, 11, 4, 1, 135, 78, 44, 26, 16, 7, 2, 176, 112, 61, 39, 23, 12, 4, 1, 231, 142, 81, 52, 32, 18, 7, 2

Offset: 0

John Tyler Rascoe, Oct 01 2023

Empirical: The first k terms of each column are A000070, for columns k > 0.

			Triangle begins:
      k=0   1  2  3  4
  n=0:  0
  n=1:  1
  n=2:  2
  n=3:  3,  1
  n=4:  5,  1
  n=5:  7,  3, 1
  n=6: 11,  3, 2
  n=7: 15,  7, 3, 1
  n=8: 22,  9, 4, 2
  n=9: 30, 15, 7, 4, 1
  ...
T(7,1) = 7: T(7,2) = 3: T(7,3) = 1:
      (43)        (331)      (4111)
     (421)       (3211)
     (322)      (31111)
    (3211)
    (2221)
   (22111)
  (211111)

John Tyler Rascoe, Python program

Cf. A000041 (column k=0), A237666 (column k=1).

Cf. A000070, A008615, A268193, A354234.

Python
```
# see linked program
```

A063277 Dimension of the space of weight n cuspidal newforms for Gamma_1( 4 ).

Original entry on oeis.org

-1, 0, 0, 0, 1, 1, 2, 0, 3, 1, 4, 1, 5, 1, 6, 1, 7, 2, 8, 1, 9, 2, 10, 2, 11, 2, 12, 2, 13, 3, 14, 2, 15, 3, 16, 3, 17, 3, 18, 3, 19, 4, 20, 3, 21, 4, 22, 4, 23, 4, 24, 4, 25, 5, 26, 4, 27, 5, 28, 5, 29, 5, 30, 5, 31, 6, 32, 5, 33, 6, 34, 6, 35, 6, 36, 6, 37, 7, 38

Offset: 2

N. J. A. Sloane, Jul 14 2001

sign

William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_1(N))
William A. Stein, The modular forms database
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 1, 0, 1, 0, 0, 0, -1).

Cf. A000027 (bisection), A008615 (bisection)

From Colin Barker, Feb 24 2016: (Start)
a(n) = a(n-4) + a(n-6) - a(n-10) for n>13.
G.f.: -x^2*(1 -2*x^4 -x^5 -3*x^6 -2*x^8) / ((1 -x)^2*(1 +x)^2*(1 -x +x^2)*(1 +x^2)*(1 +x +x^2)).
(End)

cp's OEIS Frontend

A274342 Irregular triangle read by rows, giving the numerators of the coefficients of the Eisenstein series G_{2*n} multiplied by 2*n-1, for n >= 2. Also Laurent coefficients of Weierstrass's P function.

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A340010 The order of the largest weakly connected component of the Collatz digraph of order n.

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A010764 a(n) = floor(n/2) mod floor(n/3).

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A026806 a(n) = number of numbers k such that only one partition of n has least part k.

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A274343 Irregular triangle read by rows giving the denominators of the coefficients of the Eisenstein series G_{2*n} multiplied by 2*n-1, for n >= 2. Also Laurent coefficients of Weierstrass's P function.

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A300185 Irregular triangle read by rows: T(n, {j,k}) is the number of partitions of n that have exactly j parts equal to k; 1 <= j <= n, 1 <= k <= n.

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A274342 Irregular triangle read by rows, giving the numerators of the coefficients of the Eisenstein series G_{2n} multiplied by 2n-1, for n >= 2. Also Laurent coefficients of Weierstrass's P function.

A274343 Irregular triangle read by rows giving the denominators of the coefficients of the Eisenstein series G_{2n} multiplied by 2n-1, for n >= 2. Also Laurent coefficients of Weierstrass's P function.