A014591 -id:A014591 - cp's OEIS frontend

A000598 Number of rooted ternary trees with n nodes; number of n-carbon alkyl radicals C(n)H(2n+1) ignoring stereoisomers.

1, 1, 1, 2, 4, 8, 17, 39, 89, 211, 507, 1238, 3057, 7639, 19241, 48865, 124906, 321198, 830219, 2156010, 5622109, 14715813, 38649152, 101821927, 269010485, 712566567, 1891993344, 5034704828, 13425117806, 35866550869, 95991365288, 257332864506, 690928354105

Offset: 0

N. J. A. Sloane

Number of unlabeled rooted trees in which each node has out-degree <= 3.
Ignoring stereoisomers means that the children of a node are unordered. They can be permuted in any way and it is still the same tree. See A000625 for the analogous sequence with stereoisomers counted.
In alkanes every carbon has valence exactly 4 and every hydrogen has valence exactly 1. But the trees considered here are just the carbon "skeletons" (with all the hydrogen atoms stripped off) so now each carbon bonds to 1 to 4 other carbons. The out-degree is then <= 3.
Other descriptions of this sequence: quartic planted trees with n nodes; ternary rooted trees with n nodes and height at most 3.
The number of aliphatic amino acids with n carbon atoms in the side chain, and no rings or double bonds, has the same growth as this sequence. - Konrad Gruetzmann, Aug 13 2012

			From _Joerg Arndt_, Feb 25 2017: (Start)
The a(5) = 8 rooted trees with 5 nodes and out-degrees <= 3 are:
:         level sequence    out-degrees (dots for zeros)
:     1:  [ 0 1 2 3 4 ]    [ 1 1 1 1 . ]
:  O--o--o--o--o
:
:     2:  [ 0 1 2 3 3 ]    [ 1 1 2 . . ]
:  O--o--o--o
:        .--o
:
:     3:  [ 0 1 2 3 2 ]    [ 1 2 1 . . ]
:  O--o--o--o
:     .--o
:
:     4:  [ 0 1 2 3 1 ]    [ 2 1 1 . . ]
:  O--o--o--o
:  .--o
:
:     5:  [ 0 1 2 2 2 ]    [ 1 3 . . . ]
:  O--o--o
:     .--o
:     .--o
:
:     6:  [ 0 1 2 2 1 ]    [ 2 2 . . . ]
:  O--o--o
:     .--o
:  .--o
:
:     7:  [ 0 1 2 1 2 ]    [ 2 1 . 1 . ]
:  O--o--o
:  .--o--o
:
:     8:  [ 0 1 2 1 1 ]    [ 3 1 . . . ]
:  O--o--o
:  .--o
:  .--o
(End)

N. L. Biggs et al., Graph Theory 1736-1936, Oxford, 1976, p. 62 (quoting Cayley, who is wrong).
A. Cayley, On the mathematical theory of isomers, Phil. Mag. vol. 67 (1874), 444-447 (a(6) is wrong).
J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
R. A. Fisher, Contributions to Mathematical Statistics, Wiley, 1950, 41.397.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 529.
Handbook of Combinatorics, North-Holland '95, p. 1963.
Knop, Mueller, Szymanski and Trinajstich, Computer generation of certain classes of molecules.
D. Perry, The number of structural isomers ..., J. Amer. Chem. Soc. 54 (1932), 2918-2920.
G. Polya, Mathematical and Plausible Reasoning, Vol. 1 Prob. 4 pp. 85; 233.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 (first 200 terms from N. J. A. Sloane)
Jean-François Alcover, Mathematica program translated from N. J. A. Sloane's Maple program for A000022, A000200, A000598, A000602, A000678.
A. T. Balaban, J. W. Kennedy and L. V. Quintas, The number of alkanes having n carbons and a longest chain of length d, J. Chem. Education, 65 (1988), 304-313.
A. Cayley, On the mathematical theory of isomers, Phil. Mag. vol. 67 (1874), 444-447 (a(6) is wrong). (Annotated scanned copy)
Frederic Chyzak, Enumerating alcohols and other classes of chemical molecules.
Maximilian Fichtner, K. Voigt, and S. Schuster, The tip and hidden part of the iceberg: Proteinogenic and non-proteinogenic aliphatic amino acids, Biochimica et Biophysica Acta (BBA)-General, 2016, Volume 1861, Issue 1, Part A, January 2017, pp. 3258-3269.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 478.
K. Grützmann, S. Böcker, and S. Schuster, Combinatorics of aliphatic amino acids, Naturwissenschaften, Vol. 98, No. 1, 79-86, 2011.
H. R. Henze and C. M. Blair, The number of structurally isomeric alcohols of the methanol series, J. Amer. Chem. Soc., 53 (8) (1931), 3042-3046.
H. R. Henze and C. M. Blair, The number of structurally isomeric alcohols of the methanol series, J. Amer. Chem. Soc., 53 (8) (1931), 3042-3045. (Annotated scanned copy)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1.
P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
Camden A. Parks and James B. Hendrickson, Enumeration of monocyclic and bicyclic carbon skeletons, J. Chem. Inf. Comput. Sci., vol. 31, 334-339 (1991).
D. Perry, The number of structural isomers of certain homologs of methane and methanol, J. Amer. Chem. Soc. 54 (1932), 2918-2920. [Gives a(60) correctly.] (Annotated scanned copy)
G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443; Table I, line 2.
G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443; Table I, line 2. (Annotated scanned copy)
R. C. Read, The Enumeration of Acyclic Chemical Compounds, pp. 25-61 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976. [Annotated scanned copy] See p. 20, Eq. (G); p. 27, Eq. 2.1.
Marko Riedel, Chris Grossack, Counting trees with a degree constraint, Math StackExchange.
R. W. Robinson, F. Harary and A. T. Balaban, Numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (3) (1976), 355-361.
R. W. Robinson, F. Harary and A. T. Balaban, Numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (3) (1976), 355-361. (Annotated scanned copy)
Hugo Schiff, Zur Statistik chemischer Verbindungen, Berichte der Deutschen Chemischen Gesellschaft, Vol. 8, pp. 1542-1547, 1875. [Annotated scanned copy]
N. J. A. Sloane, Maple program and first 60 terms for A000022, A000200, A000598, A000602, A000678.
N. Trinajstich, Z. Jerievi, J. V. Knop, W. R. Muller and K. Szymanski, Computer Generation of Isomeric Structures, Pure & Appl. Chem., Vol. 55, No. 2, pp. 379-390, 1983.
Wikipedia, Polya's enumeration theorem.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A000599, A000600, A000602, A000625, A000628, A000678, A010372, A010373, A086194, A086200, A261340.
Cf. A292553, A292554, A292555, A292556.
Cf. A000081, A001190, A014591, A032305, A295461, A298118, A298120, A298204, A298422, A298426.
Column k=3 of A299038.

N := 45; G000598 := 0: i := 0: while i<(N+1) do G000598 := series(1+z*(G000598^3/6+subs(z=z^2,G000598)*G000598/2+subs(z=z^3,G000598)/3)+O(z^(N+1)),z,N+1): t[ i ] := G000598: i := i+1: od: A000598 := n->coeff(G000598,z,n);
# Another Maple program for g.f. G000598:
G000598 := 1; f := proc(n) global G000598; coeff(series(1+(1/6)*x*(G000598^3+3*G000598*subs(x=x^2,G000598)+2*subs(x=x^3,G000598)),x, n+1),x,n); end; for n from 1 to 50 do G000598 := series(G000598+f(n)*x^n,x,n+1); od; G000598;
spec := [S, {Z=Atom, S=Union(Z, Prod(Z, Set(S, card=3)))}, unlabeled]: [seq(combstruct[count](spec, size=n), n=0..20)];

m = 45; Clear[f]; f[1, x_] := 1+x; f[n_, x_] := f[n, x] = Expand[1+x*(f[n-1, x]^3/6 + f[n-1, x^2]*f[n-1, x]/2 + f[n-1, x^3]/3)][[1 ;; n]]; Do[f[n, x], {n, 2, m}]; CoefficientList[f[m, x], x]
(* second program (after N. J. A. Sloane): *)
m = 45; gf[] = 0; Do[gf[z] = 1 + z*(gf[z]^3/6 + gf[z^2]*gf[z]/2 + gf[z^3]/3) + O[z]^m // Normal, m]; CoefficientList[gf[z], z]  (* Jean-François Alcover, Sep 23 2014, updated Jan 11 2018 *)
b[0, i_, t_, k_] = 1; m = 3; (* m = maximum children *)
b[n_,i_,t_,k_]:= b[n,i,t,k]= If[i<1,0,
  Sum[Binomial[b[i-1, i-1, k, k] + j-1, j]*
  b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]];
Join[{1},Table[b[n-1, n-1, m, m], {n, 1, 35}]] (* Robert A. Russell, Dec 27 2022 *)

seq(n)={my(g=O(x)); for(n=1, n, g = 1 + x*(g^3/6 + subst(g,x,x^2)*g/2 + subst(g,x,x^3)/3) + O(x^n)); Vec(g)} \\ Andrew Howroyd, May 22 2018

def seq(n):
    B = PolynomialRing(QQ, 't', n+1);t = B.gens()
    R. = B[[]]
    T = sum([t[i] * z^i for i in range(1,n+1)]) + O(z^(n+1))
    lhs, rhs = T, 1 + z/6 * (T(z)^3 + 3*T(z)*T(z^2) + 2*T(z^3))
    I = B.ideal([lhs.coefficients()[i] - rhs.coefficients()[i] for i in range(n)])
    return [I.reduce(t[i]) for i in range(1,n+1)]
seq(33) # Chris Grossack, Mar 31 2025

G.f. A(x) satisfies A(x) = 1 + (1/6)*x*(A(x)^3 + 3*A(x)*A(x^2) + 2*A(x^3)).

a(n) ~ c * d^n / n^(3/2), where d = 1/A261340 = 2.8154600331761507465266167782426995425365065396907..., c = 0.517875906458893536993162356992854345458168348098... . - Vaclav Kotesovec, Aug 15 2015

Additional comments from Steve Strand (snstrand(AT)comcast.net), Aug 20 2003

A298204 Number of unlabeled rooted trees with n nodes in which all outdegrees are either 0, 1, or 3.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 16, 29, 55, 104, 200, 389, 763, 1507, 3002, 6010, 12102, 24484, 49751, 101475, 207723, 426542, 878451, 1813945, 3754918, 7790326, 16196629, 33739335, 70410401, 147187513, 308171861, 646188276, 1356847388, 2852809425, 6005542176

Offset: 1

Gus Wiseman, Jan 14 2018

nonn

			The a(7) = 9 trees: ((((((o)))))), ((((ooo)))), (((oo(o)))), ((oo((o)))), ((o(o)(o))), (oo(((o)))), (oo(ooo)), (o(o)((o))), ((o)(o)(o)).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Gus Wiseman, Illustration of the first 8 terms.

Cf. A000081, A000598, A001190, A001399, A004111, A014591, A032305, A273873, A292050, A295461, A298118, A298205.

b:= proc(n, i, v) option remember; `if`(n=0,
      `if`(v=0, 1, 0), `if`(i<1 or v<1 or n `if`(n<2, n, add(b(n-1$2, j), j=[1, 3])):
seq(a(n), n=1..40);  # Alois P. Heinz, Jan 30 2018

multing[n_,k_]:=Binomial[n+k-1,k];
a[n_]:=a[n]=If[n===1,1,Sum[Product[multing[a[x],Count[ptn,x]],{x,Union[ptn]}],{ptn,Select[IntegerPartitions[n-1],MemberQ[{1,3},Length[#]]&]}]];
Table[a[n],{n,40}]
(* Second program: *)
b[n_, i_, v_] := b[n, i, v] = If[n == 0,
     If[v == 0, 1, 0], If[i < 1 || v < 1 || n < v, 0,
     If[n == v, 1, Sum[Binomial[a[i] + j - 1, j]*
     b[n - i*j, i - 1, v - j], {j, 0, Min[n/i, v]}]]]];
a[n_] := If[n < 2, n, Sum[b[n - 1, n - 1, j], {j, {1, 3}}]];
Array[a, 40] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

A274921 Spiral constructed on the nodes of the triangular net in which each new term is the least positive integer distinct from its neighbors.

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 3, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 1, 2, 1, 3, 2, 3, 1, 2, 3, 2, 1, 3, 2, 3, 1, 2, 3, 2, 1, 3, 2, 3, 1, 2, 3, 1, 3, 2, 1, 3, 1, 2, 3, 1, 2, 1, 3, 2, 1, 3, 1, 2, 3, 1, 2, 1, 3, 2, 1, 3, 1, 2, 3, 1, 2, 3, 2, 1, 3, 2, 1, 2, 3, 1, 2, 3, 1, 3, 2, 1

Offset: 0

Omar E. Pol, Jul 11 2016

nonn

The structure of the spiral has the following properties:
1) Every 1 is surrounded by three equidistant 2's and three equidistant 3's.
2) Every 2 is surrounded by three equidistant 1's and three equidistant 3's.
3) Every 3 is surrounded by three equidistant 1's and three equidistant 2's.
4) Diagonals are periodic sequences with period 3 (A010882 and A130784).
From Juan Pablo Herrera P., Nov 16 2016: (Start)
5) Every hexagon with a 1 in its center is the same hexagon as the one in the middle of the spiral.
6) Every triangle whose number of numbers is divisible by 3 has the same number of 1's, 2's, and 3's. For example, a triangle with 6 numbers, has two 1's, two 2's, and two 3's. (End)
a(n) = a(n-2) if n > 2 is in A014591, otherwise a(n) = 6 - a(n-1)-a(n-2). - Robert Israel, Sep 15 2017

			Illustration of initial terms as a spiral:
.
.                3 - 1 - 2 - 3 - 1 - 2
.               /                     \
.              1   2 - 3 - 1 - 2 - 3   1
.             /   /                 \   \
.            2   3   1 - 2 - 3 - 1   2   3
.           /   /   /             \   \   \
.          3   1   2   3 - 1 - 2   3   1   2
.         /   /   /   /         \   \   \   \
.        1   2   3   1   2 - 3   1   2   3   1
.       /   /   /   /   /     \   \   \   \   \
.      2   3   1   2   3   1 - 2   3   1   2   3
.       \   \   \   \   \         /   /   /   /
.        1   2   3   1   2 - 3 - 1   2   3   1
.         \   \   \   \             /   /   /
.          3   1   2   3 - 1 - 2 - 3   1   2
.           \   \   \                 /   /
.            2   3   1 - 2 - 3 - 1 - 2   3
.             \   \                     /
.              1   2 - 3 - 1 - 2 - 3 - 1
.               \
.                3 - 1 - 2 - 3 - 1 - 2
.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A001399, A010882, A130784, A253186, A274820, A274821, A274920, A275606, A275610.

A[0]:= 1: A[1]:= 2: A[2]:= 3:
b:= 3: c:= 2: d:= 2: e:= 1: f:= 1:
for n from 3 to 200 do
  if n = b then
     r:= b; b:= c + d - f + 1; f:= e; e:= d; d:= c; c:= r;
     A[n]:= A[n-2];
  else
     A[n]:= 6 - A[n-1] - A[n-2];
  fi
od:
seq(A[i],i=0..200); # Robert Israel, Sep 15 2017

a(n) = A274920(n) + 1.

A347586 Number of partitions of n into at most 4 distinct parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 26, 31, 36, 43, 49, 57, 65, 75, 84, 96, 107, 121, 134, 150, 165, 184, 201, 222, 242, 266, 288, 315, 340, 370, 398, 431, 462, 499, 533, 573, 611, 655, 696, 744, 789, 841, 890, 946, 999, 1060, 1117, 1182, 1244, 1314, 1380, 1455

Offset: 0

Ilya Gutkovskiy, Sep 08 2021

Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-2,0,0,1,1,-1).

Cf. A000578, A001400, A014591, A065033, A347587, A347588.

nmax = 60; CoefficientList[Series[Sum[x^(k (k + 1)/2)/Product[(1 - x^j), {j, 1, k}], {k, 0, 4}], {x, 0, nmax}], x]
Join[{1}, LinearRecurrence[{1, 1, 0, 0, -2, 0, 0, 1, 1, -1}, {1, 1, 2, 2, 3, 4, 5, 6, 8, 10}, 60]]

G.f.: Sum_{k=0..4} x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j).

a(n) ~ A000578(n)/144. - Stefano Spezia, Sep 08 2021

A347549 Number of partitions of n into 4 or more distinct parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 30, 38, 48, 59, 73, 89, 108, 130, 156, 185, 220, 259, 304, 356, 415, 482, 559, 645, 743, 854, 979, 1119, 1278, 1455, 1654, 1878, 2127, 2405, 2717, 3063, 3449, 3879, 4356, 4885, 5474, 6125, 6846, 7645, 8527, 9501, 10579

Offset: 10

Ilya Gutkovskiy, Sep 06 2021

nonn

Cf. A000009, A014591, A111133, A347548.

nmax = 60; CoefficientList[Series[Sum[x^(k (k + 1)/2)/Product[(1 - x^j), {j, 1, k}], {k, 4, nmax}], {x, 0, nmax}], x] // Drop[#, 10] &

G.f.: Sum_{k>=4} x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j).

a(n) = A000009(n) - A014591(n). - Vaclav Kotesovec, Sep 14 2021

A298207 Numbers that are a product of zero, one, or three (not necessarily distinct) prime numbers.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 20, 23, 27, 28, 29, 30, 31, 37, 41, 42, 43, 44, 45, 47, 50, 52, 53, 59, 61, 63, 66, 67, 68, 70, 71, 73, 75, 76, 78, 79, 83, 89, 92, 97, 98, 99, 101, 102, 103, 105, 107, 109, 110, 113, 114, 116, 117, 124, 125, 127, 130

Offset: 1

Gus Wiseman, Jan 14 2018

nonn

			1 is a product of zero primes so is in the sequence.
6 = 2 * 3 is a product of two primes so is not in the sequence.
12 = 2 * 2 * 3 is a product of three primes so is in the sequence.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000040, A001358, A001399, A008578, A014591, A014612, A026424, A028260, A298120, A298204, A298205.

Subsequence of A037144. - Felix Fröhlich, Jan 15 2018

a:= proc(n) option remember; local k; for k
      from 1+`if`(n=1, 0, a(n-1)) while not
      numtheory[bigomega](k) in {0, 1, 3} do od; k
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Jan 15 2018

Select[Range[200],MemberQ[{0,1,3},PrimeOmega[#]]&]

is(n) = my(v=[0, 1, 3]); #setintersect([bigomega(n)], v)==1 \\ Felix Fröhlich, Jan 15 2018

Equals {A008578} union {A014612}.

Equals {A037144} minus {A001358}.

A347587 Number of partitions of n into at most 5 distinct parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, 46, 54, 64, 75, 88, 102, 119, 137, 158, 181, 207, 235, 268, 302, 341, 383, 430, 480, 536, 595, 661, 731, 808, 889, 979, 1073, 1176, 1285, 1403, 1527, 1662, 1803, 1956, 2116, 2288, 2468, 2662, 2864, 3080, 3306, 3547

Offset: 0

Ilya Gutkovskiy, Sep 08 2021

Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,-1,-1,1,1,1,0,0,-1,-1,1).

Cf. A001401, A014591, A065033, A347586, A347588.

nmax = 58; CoefficientList[Series[Sum[x^(k (k + 1)/2)/Product[(1 - x^j), {j, 1, k}], {k, 0, 5}], {x, 0, nmax}], x]
LinearRecurrence[{1, 1, 0, 0, -1, -1, -1, 1, 1, 1, 0, 0, -1, -1, 1}, {1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22}, 59]

G.f.: Sum_{k=0..5} x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j).

A347588 Number of partitions of n into at most 6 distinct parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, 46, 54, 64, 76, 89, 104, 122, 142, 165, 192, 221, 255, 294, 337, 385, 441, 501, 570, 646, 731, 824, 930, 1043, 1171, 1310, 1464, 1630, 1817, 2015, 2236, 2473, 2734, 3013, 3322, 3648, 4008, 4391, 4809, 5252, 5738, 6249

Offset: 0

Ilya Gutkovskiy, Sep 08 2021

Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,0,-2,0,1,1,1,1,0,-2,0,-1,0,0,1,1,-1).

Cf. A001402, A014591, A065033, A347586, A347587.

nmax = 58; CoefficientList[Series[Sum[x^(k (k + 1)/2)/Product[(1 - x^j), {j, 1, k}], {k, 0, 6}], {x, 0, nmax}], x]
Join[{1}, LinearRecurrence[{1, 1, 0, 0, -1, 0, -2, 0, 1, 1, 1, 1, 0, -2, 0, -1, 0, 0, 1, 1, -1}, {1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, 46, 54, 64, 76}, 58]]

G.f.: Sum_{k=0..6} x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j).

A298205 Matula-Goebel numbers of rooted trees in which all outdegrees are either 0, 1, or 3.

Original entry on oeis.org

1, 2, 3, 5, 8, 11, 12, 18, 19, 20, 27, 30, 31, 37, 44, 45, 50, 61, 66, 67, 71, 75, 76, 99, 103, 110, 113, 114, 124, 125, 127, 148, 157, 165, 171, 186, 190, 193, 197, 222, 229, 242, 244, 268, 275, 279, 283, 284, 285, 310, 317, 331, 333, 353, 363, 366, 370, 379

Offset: 1

Gus Wiseman, Jan 14 2018

nonn

			Sequence of rooted trees begins:
1  o
2  (o)
3  ((o))
5  (((o)))
8  (ooo)
11 ((((o))))
12 (oo(o))
18 (o(o)(o))
19 ((ooo))
20 (oo((o)))
27 ((o)(o)(o))
30 (o(o)((o)))
31 (((((o)))))
37 ((oo(o)))
44 (oo(((o))))
45 ((o)(o)((o)))
50 (o((o))((o)))

Cf. A000081, A000598, A014591, A026424, A032305, A061775, A111299, A276625, A292050, A295461, A297571, A298126, A298118, A298120, A298204, A298207.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stQ[n_]:=Or[n===1,With[{m=primeMS[n]},MemberQ[{1,3},Length[m]]&&And@@stQ/@m]];
Select[Range[10000],stQ]

cp's OEIS Frontend

A000598 Number of rooted ternary trees with n nodes; number of n-carbon alkyl radicals C(n)H(2n+1) ignoring stereoisomers.

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A298204 Number of unlabeled rooted trees with n nodes in which all outdegrees are either 0, 1, or 3.

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A274921 Spiral constructed on the nodes of the triangular net in which each new term is the least positive integer distinct from its neighbors.

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A347586 Number of partitions of n into at most 4 distinct parts.

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A347549 Number of partitions of n into 4 or more distinct parts.

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A298207 Numbers that are a product of zero, one, or three (not necessarily distinct) prime numbers.

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A347587 Number of partitions of n into at most 5 distinct parts.

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A347588 Number of partitions of n into at most 6 distinct parts.

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