A024718 -id:A024718 - cp's OEIS frontend

A384960 a(n) = smallest sphenic number k such that A010846(k) = n.

1001, 105, 231, 30, 42, 70, 110, 66, 78, 170, 102, 114, 138, 370, 174, 826, 222, 246, 258, 318, 354, 402, 438, 498, 534, 582, 654, 762, 786, 894, 978, 1038, 1158, 1338, 1506, 1542, 1758, 1986, 2082, 2202, 2334, 2598, 2922, 3126, 3462, 3918, 4098, 4398, 4614, 5262

Offset: 15

Michael De Vlieger, Jul 06 2025

nonn

a(1) = A384000(3) = 1001; A010846(1001) = A024718(3) = 15; 1001 is the smallest number k with 3 distinct prime factors that has the smallest possible number of terms in row k of A162306, i.e., m <= k such that rad(m) | k.

For n > 30, 6 | a(n).

			Table of a(n) indicating prime factors and S, where S = {ceiling(log_p a(n))} for all primes p that divide a(n), in order of the magnitude of p.
                                Prime power factor
                                    1111223344455
 n  m=a(n) pi(facs(m))    S     23571379391713739
-------------------------------------------------
15   1001   4.5.6       4.3.3   ...111
16    105   2.3.4       5.3.3   .111
17    231   2.4.5       5.3.3   .1.11
18     30   1.2.3       5.4.3   111
19     42   1.2.4       6.4.2   11.1
20     70   1.3.4       7.3.3   1.11
21    110   1.3.5       7.3.2   1.1.1
22     66   1.2.5       7.4.2   11..1
23     78   1.2.6       7.4.2   11...1
24    170   1.3.7       8.4.2   1.1...1
25    102   1.2.7       7.5.2   11....1
26    114   1.2.8       7.5.2   11.....1
27    138   1.2.9       8.5.2   11......1
28    370   1.3.12      9.4.2   1.1........1
29    174   1.2.10      8.5.2   11.......1
30    826   1.4.17     10.4.2   1..1............1
31    222   1.2.12      8.5.2   11.........1
32    246   1.2.13      8.6.2   11..........1
33    258   1.2.14      9.6.2   11...........1
34    318   1.2.16      9.6.2   11.............1

Michael De Vlieger, Table of n, a(n) for n = 15..300
Michael De Vlieger, Plot of terms k = p^a*q^b*r^c, primes p < q < r, in row a(n) of A162306, n = 15..50, at (x,y,z) = (a,b,c). For a(n) there are n blocks in each diagram.
Michael De Vlieger, Mathematica code.

Cf. A007304, A007947, A010846, A024718, A162306, A384000.

(* See Mathematica code link for function definitions for kappaomega and theta *)
s =  kappaomega[3, 6000]; t = Map[theta, s];
Map[s[[FirstPosition[t, #][[1]] ]] &, Union[t]]

A114502 Triangle read by rows: T(n,k) is number of ordered trees with n edges and having exactly k vertices all of whose children are leaves (1<=k<=floor(n/2) for n>=2).

Original entry on oeis.org

1, 2, 5, 13, 1, 34, 8, 89, 42, 1, 233, 183, 13, 610, 717, 102, 1, 1597, 2622, 624, 19, 4181, 9134, 3275, 205, 1, 10946, 30691, 15473, 1650, 26, 28657, 100284, 67684, 11020, 366, 1, 75025, 320466, 279106, 64553, 3716, 34, 196418, 1005630, 1098402, 342867

Offset: 1

Emeric Deutsch, Dec 02 2005

Row 1 has one term; row n (n>=2) has floor(n/2) terms. Row sums are the Catalan numbers (A000108). Column 1 yields the Fibonacci numbers with odd index (A001519). Sum(kT(n,k),k=1..floor(n/2))=[1+sum(binomial(2j,j),j=0..n-1)]/2 (A024718).

			T(4,2)=1 because we have the tree with two paths of length two, rab and rcd, emanating from the root r; a and b are vertices all of whose children are leaves.
Triangle starts:
  1;
  2;
  5;
  13,1;
  34,8;
  89,42,1;
  233,183,13;
  610,717,102,1;
  ...

Jean-Luc Baril, Pamela E. Harris, Kimberly J. Harry, Matt McClinton, and José L. Ramírez, Enumerating runs, valleys, and peaks in Catalan words, arXiv:2404.05672 [math.CO], 2024. See p. 14.

Cf. A000108, A001519, A024718.

G:=1/2/(z^2-z)*(-1+z+z^2-t*z^2+sqrt(1-6*z+11*z^2-2*t*z^2-6*z^3+2*z^3*t+z^4-2*z^4*t+t^2*z^4)): Gser:=simplify(series(G,z=0,18)): for n from 1 to 15 do P[n]:=coeff(Gser,z^n) od: 1; for n from 1 to 15 do seq(coeff(P[n],t^j),j=1..floor(n/2)) od; # yields sequence in triangular form

G.f.: G=G(t, z) satisfies z(1-z)G^2-(1-z-z^2+tz^2)G+1-2z+tz=0.

G.f. G(t,z) can be derived easily from the symbolic decomposition of an ordered tree according to the degree of the root; one obtains G = 1 + z*(G-1+t) + z^2*(G^2-1+t) + z^3*(G^3-1+t) + ... . - Emeric Deutsch, Feb 12 2015

A171670 Triangle T read by rows : T(n,k)= A007318(n,k)*A005773(n-k).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 6, 3, 1, 13, 20, 12, 4, 1, 35, 65, 50, 20, 5, 1, 96, 210, 195, 100, 30, 6, 1, 267, 672, 735, 455, 175, 42, 7, 1, 750, 2136, 2688, 1960, 910, 280, 56, 8, 1, 2123, 6750, 9612, 8064, 4410, 1638, 420, 72, 9, 1

Offset: 0

Philippe Deléham, Dec 14 2009

			Triangle begins : 1 ; 1,1 ; 2,2,1 ; 5,6,3,1 ; 13,20,12,4,1 ; 35,65,50,20,5,1 ; ...

Cf. A171651

Sum_{k, 0<=k<=n} T(n,k)= A024718(n).

A361802 Irregular triangle read by rows where T(n,k) is the number of k-subsets of {-n+1,...,n} with sum 0, for k = 1,...,2n-1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 3, 6, 7, 5, 2, 1, 1, 4, 10, 16, 18, 14, 8, 3, 1, 1, 5, 15, 31, 46, 51, 43, 27, 12, 3, 1, 1, 6, 21, 53, 98, 139, 155, 134, 88, 43, 16, 4, 1, 1, 7, 28, 83, 184, 319, 441, 486, 424, 293, 161, 68, 21, 4, 1

Offset: 1

Gus Wiseman, Apr 10 2023

Also the number of k-subsets of {1,...,2n} with mean n.

			Triangle begins:
   1
   1   1   1
   1   2   3   2   1
   1   3   6   7   5   2   1
   1   4  10  16  18  14   8   3   1
   1   5  15  31  46  51  43  27  12   3   1
   1   6  21  53  98 139 155 134  88  43  16   4   1
   1   7  28  83 184 319 441 486 424 293 161  68  21   4   1
Row n = 3 counts the following subsets:
  {0}  {-1,1}  {-1,0,1}   {-2,-1,0,3}  {-2,-1,0,1,2}
       {-2,2}  {-2,0,2}   {-2,-1,1,2}
               {-2,-1,3}

Row lengths are A005408.
Row sums are A212352.
A007318 counts subsets by length.
A067538 counts partitions with integer mean.
A231147 counts subsets by median.
A327475 counts subsets with integer mean, median A000975.
A327481 counts subsets by mean.
Cf. A006134, A013580, A024718, A079309, A326512, A349156, A361654, A362046.

Table[Length[Select[Subsets[Range[-n+1,n],{k}],Total[#]==0&]],{n,6},{k,2n-1}]

cp's OEIS Frontend

A384960 a(n) = smallest sphenic number k such that A010846(k) = n.

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A114502 Triangle read by rows: T(n,k) is number of ordered trees with n edges and having exactly k vertices all of whose children are leaves (1<=k<=floor(n/2) for n>=2).

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A171670 Triangle T read by rows : T(n,k)= A007318(n,k)*A005773(n-k).

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Formula

A361802 Irregular triangle read by rows where T(n,k) is the number of k-subsets of {-n+1,...,n} with sum 0, for k = 1,...,2n-1.

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Examples

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Programs

Mathematica