A290822 -id:A290822 - cp's OEIS frontend

A324742 Number of subsets of {2...n} containing no prime indices of the elements.

1, 2, 3, 6, 10, 16, 24, 48, 84, 144, 228, 420, 648, 1080, 1800, 3600, 5760, 11136, 16704, 31104, 53568, 90624, 136896, 269952, 515712, 862080, 1708800, 3171840, 4832640, 9325440, 14890752, 29781504, 52245504, 88418304, 166017024, 331628544, 497645568, 829409280

Offset: 1

Gus Wiseman, Mar 15 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(1) = 1 through a(6) = 16 subsets:
  {}  {}   {}   {}     {}       {}
      {2}  {2}  {2}    {2}      {2}
           {3}  {3}    {3}      {3}
                {4}    {4}      {4}
                {2,4}  {5}      {5}
                {3,4}  {2,4}    {6}
                       {2,5}    {2,4}
                       {3,4}    {2,5}
                       {4,5}    {3,4}
                       {2,4,5}  {3,6}
                                {4,5}
                                {4,6}
                                {5,6}
                                {2,4,5}
                                {3,4,6}
                                {4,5,6}
An example for n = 20 is {4,5,6,12,17,18,19}, with prime indices:
   4: {1,1}
   5: {3}
   6: {1,2}
  12: {1,1,2}
  17: {7}
  18: {1,2,2}
  19: {8}
None of these prime indices {1,2,3,7,8} belong to the set, as required.

Andrew Howroyd, Table of n, a(n) for n = 1..100

The maximal case is A324763. The version for subsets of {1...n} is A324741. The strict integer partition version is A324752. The integer partition version is A324757. The Heinz number version is A324761. An infinite version is A304360.
Cf. A000720, A001462, A007097, A076078, A084422, A085945, A112798, A276625, A290689, A290822, A306844, A324764.
Cf. A324695, A324737, A324743, A324751, A324756, A324758.

Table[Length[Select[Subsets[Range[2,n]],Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]],{n,10}]

pset(n)={my(b=0,f=factor(n)[,1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n-1,k,pset(k+1)>>1), d=0); for(i=1, #p, d=bitor(d, p[i]));
((k,b)->if(k>#p, 1, my(t=self()(k+1,b)); if(!bitand(p[k], b), t+=if(bittest(d,k), self()(k+1, b+(1<Andrew Howroyd, Aug 16 2019

Terms a(21) and beyond from Andrew Howroyd, Aug 16 2019

A352488 Weak nonexcedance set of A122111. Numbers k >= A122111(k), where A122111 represents partition conjugation using Heinz numbers.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 16, 18, 20, 24, 27, 30, 32, 36, 40, 48, 50, 54, 56, 60, 64, 72, 75, 80, 81, 84, 90, 96, 100, 108, 112, 120, 125, 128, 135, 140, 144, 150, 160, 162, 168, 176, 180, 192, 196, 200, 210, 216, 224, 225, 240, 243, 250, 252, 256, 264, 270, 280

Offset: 1

Gus Wiseman, Mar 20 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The sequence lists all Heinz numbers of partitions whose Heinz number is greater than or equal to that of their conjugate.

			The terms together with their prime indices begin:
    1: ()
    2: (1)
    4: (1,1)
    6: (2,1)
    8: (1,1,1)
    9: (2,2)
   12: (2,1,1)
   16: (1,1,1,1)
   18: (2,2,1)
   20: (3,1,1)
   24: (2,1,1,1)
   27: (2,2,2)
   30: (3,2,1)
   32: (1,1,1,1,1)
   36: (2,2,1,1)
   40: (3,1,1,1)
   48: (2,1,1,1,1)
   50: (3,3,1)
   54: (2,2,2,1)
   56: (4,1,1,1)

Alois P. Heinz, Table of n, a(n) for n = 1..2000
Richard Ehrenborg and Einar Steingrímsson, The Excedance Set of a Permutation, Advances in Applied Mathematics 24, (2000), 284-299.
MathOverflow, Why 'excedances' of permutations? [closed].

These partitions are counted by A046682.
The opposite version is A352489, strong A352487.
The strong version is A352490, counted by A000701.
These are the positions of nonnegative terms in A352491.
A000041 counts integer partitions, strict A000009.
A000700 counts self-conjugate partitions, ranked by A088902 (cf. A258116).
A003963 = product of prime indices, conjugate A329382.
A008292 is the triangle of Eulerian numbers (version without zeros).
A008480 counts permutations of prime indices, conjugate A321648.
A056239 adds up prime indices, row sums of A112798 and A296150.
A122111 = partition conjugation using Heinz numbers, parts A321649/A321650.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A173018 counts permutations by excedances, weak A123125.
A330644 counts non-self-conjugate partitions, ranked by A352486.
A352525 counts compositions by weak superdiagonals, rank statistic A352517.
Cf. A000720, A045931, A114088, A120383, A175508, A290822, A319005, A325040, A325698, A347450.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],#>=Times@@Prime/@conj[primeMS[#]]&]

a(n) >= A122111(a(n)).

A324843 Number of unlabeled rooted trees with n nodes where the branches of any branch of any terminal subtree form a submultiset of the branches of the same subtree.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 8, 9, 15, 17, 31, 35, 57, 70, 111, 136, 213, 265, 405, 517, 763, 987, 1458, 1893, 2736, 3611, 5161, 6836, 9702

Offset: 1

Gus Wiseman, Mar 18 2019

A subset of totally transitive rooted trees (A318185).

			The a(1) = 1 through a(8) = 8 rooted trees:
  o  (o)  (oo)  (ooo)   (oooo)   (ooooo)    (oooooo)    (ooooooo)
                (o(o))  (oo(o))  (oo(oo))   (ooo(oo))   (ooo(ooo))
                                 (ooo(o))   (oooo(o))   (oooo(oo))
                                 (o(o)(o))  (oo(o)(o))  (ooooo(o))
                                                        (oo(o)(oo))
                                                        (ooo(o)(o))
                                                        (o(o)(o)(o))
                                                        (o(o)(o(o)))

The Matula-Goebel numbers of these trees are given by A324842.
Cf. A000081, A279861, A290689, A290822, A318185.
Cf. A324704, A324736, A324748, A324753, A324847, A324848, A324854.

submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];
rallt[n_]:=Select[Union[Sort/@Join@@(Tuples[rallt/@#]&/@IntegerPartitions[n-1])],And@@Table[submultQ[b,#],{b,#}]&];
Table[Length[rallt[n]],{n,10}]

A324935 Matula-Goebel numbers of rooted trees whose non-leaf terminal subtrees are all different.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 19, 20, 21, 22, 24, 26, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 42, 43, 44, 48, 51, 52, 53, 56, 57, 58, 59, 62, 64, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 84, 85, 86, 88, 89, 91, 95, 96, 101, 102, 104

Offset: 1

Gus Wiseman, Mar 21 2019

nonn

Every positive integer has a unique factorization into factors q(i) = prime(i)/i, i > 0. This sequence consists of all numbers where this factorization has all distinct factors, except possibly for any multiplicity of q(1). For example, 22 = q(1)^2 q(2) q(3) q(5) is in the sequence, while 50 = q(1)^3 q(2)^2 q(3)^2 is not.

The enumeration of these trees by number of vertices is A324936.

			The sequence of trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   6: (o(o))
   7: ((oo))
   8: (ooo)
  10: (o((o)))
  11: ((((o))))
  12: (oo(o))
  13: ((o(o)))
  14: (o(oo))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  20: (oo((o)))
  21: ((o)(oo))
  22: (o(((o))))
  24: (ooo(o))
  26: (o(o(o)))
  28: (oo(oo))
  29: ((o((o))))
  31: (((((o)))))

Gus Wiseman, The first 36 rooted trees whose non-leaf terminal subtrees are all different, together with their Matula-Goebel numbers.

Cf. A000081, A004111, A007097, A061775, A196050, A276625, A290822, A317713.

Cf. A324850, A324922, A324923, A324924, A324931, A324934, A324936.

difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Select[Range[100],UnsameQ@@DeleteCases[difac[#],1]&]

A301345 Regular triangle where T(n,k) is the number of transitive rooted trees with n nodes and k leaves.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 1, 2, 4, 1, 0, 0, 0, 0, 3, 4, 5, 1, 0, 0, 0, 0, 2, 6, 6, 6, 1, 0, 0, 0, 0, 1, 6, 10, 9, 7, 1, 0, 0, 0, 0, 1, 5, 12, 16, 12, 8, 1, 0, 0, 0, 0, 0, 4, 13, 22, 23, 16, 9, 1, 0, 0, 0, 0, 0, 3, 14, 27, 36, 32, 20, 10, 1, 0, 0, 0, 0, 0, 2, 11

Offset: 1

Gus Wiseman, Mar 19 2018

			Triangle begins:
1
1   0
0   1   0
0   1   1   0
0   0   2   1   0
0   0   1   3   1   0
0   0   1   2   4   1   0
0   0   0   3   4   5   1   0
0   0   0   2   6   6   6   1   0
0   0   0   1   6  10   9   7   1   0
0   0   0   1   5  12  16  12   8   1   0
The T(9,5) = 6 transitive rooted trees: (o(o)(oo(o))), (o((oo))(oo)), (oo(o)(o(o))), (o(o)(o)(oo)), (ooo(o)((o))), (oo(o)(o)(o)).

Row sums are A290689.

Cf. A000081, A001190, A003238, A004111, A032305, A055277, A276625, A279861, A290760, A290822, A298422, A298426, A301342, A301343, A301344.

rut[n_]:=rut[n]=If[n===1,{{}},Join@@Function[c,Union[Sort/@Tuples[rut/@c]]]/@IntegerPartitions[n-1]];
trat[n_]:=Select[rut[n],Complement[Union@@#,#]==={}&];
Table[Length[Select[trat[n],Count[#,{},{-2}]===k&]],{n,15},{k,n}]

A324748 Number of strict integer partitions of n containing all prime indices of the parts.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 2, 3, 2, 2, 4, 3, 4, 3, 5, 6, 9, 8, 7, 8, 11, 12, 13, 15, 17, 22, 22, 20, 28, 31, 32, 36, 41, 43, 53, 53, 59, 70, 76, 77, 89, 99, 108, 124, 135, 139, 160, 172, 188, 209, 229, 243, 274, 298, 315, 353, 391, 417, 457, 496, 538, 588

Offset: 0

Gus Wiseman, Mar 15 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The first 15 terms count the following integer partitions.
   1: (1)
   3: (2,1)
   5: (4,1)
   6: (3,2,1)
   7: (4,2,1)
   9: (8,1)
   9: (6,2,1)
  10: (4,3,2,1)
  11: (8,2,1)
  11: (5,3,2,1)
  12: (9,2,1)
  12: (7,4,1)
  12: (6,3,2,1)
  13: (8,4,1)
  13: (6,4,2,1)
  14: (8,3,2,1)
  14: (7,4,2,1)
  15: (12,2,1)
  15: (9,3,2,1)
  15: (8,4,2,1)
  15: (5,4,3,2,1)
An example for n = 6 is (20,18,11,5,3,2,1), with prime indices:
  20: {1,1,3}
  18: {1,2,2}
  11: {5}
   5: {3}
   3: {2}
   2: {1}
   1: {}
All of these prime indices {1,2,3,5} belong to the partition, as required.

The subset version is A324736. The non-strict version is A324753. The Heinz number version is A290822. An infinite version is A324698.
Cf. A000720, A001462, A007097, A074971, A078374, A112798, A276625, A279861, A290689, A290760, A305713.
Cf. A324697, A324737, A324751.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&SubsetQ[#,PrimePi/@First/@Join@@FactorInteger/@DeleteCases[#,1]]&]],{n,0,30}]

A358453 Number of transitive ordered rooted trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 17, 37, 83, 190, 444, 1051, 2518, 6090, 14852

Offset: 1

Gus Wiseman, Nov 18 2022

We define an unlabeled ordered rooted tree to be transitive if every branch of a branch of the root already appears farther to the left as a branch of the root. An undirected version is A358454.

			The a(1) = 1 through a(7) = 17 trees:
  o  (o)  (oo)  (ooo)   (oooo)   (ooooo)    (oooooo)
                (o(o))  (o(o)o)  (o(o)oo)   (o(o)ooo)
                        (o(oo))  (o(oo)o)   (o(oo)oo)
                        (oo(o))  (o(ooo))   (o(ooo)o)
                                 (oo(o)o)   (o(oooo))
                                 (oo(oo))   (oo(o)oo)
                                 (ooo(o))   (oo(oo)o)
                                 (o(o)(o))  (oo(ooo))
                                            (ooo(o)o)
                                            (ooo(oo))
                                            (oooo(o))
                                            (o(o)(o)o)
                                            (o(o)(oo))
                                            (o(o)o(o))
                                            (o(oo)(o))
                                            (oo(o)(o))
                                            (o(o)((o)))

The unordered version is A290689, ranked by A290822.
The undirected version is A358454, ranked by A358458.
These trees are ranked by A358457.
A000081 counts rooted trees.
A306844 counts anti-transitive rooted trees.
Cf. A318185, A324695, A324751, A324756, A324758, A324764-A324768, A324838, A324840, A324844, A358456.

aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],Function[t,And@@Table[Complement[t[[k]],Take[t,k]]=={},{k,Length[t]}]]]],{n,10}]

A184155 The Matula-Goebel number of rooted trees having all leaves at the same level.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 17, 19, 21, 23, 25, 27, 31, 32, 49, 53, 57, 59, 63, 64, 67, 73, 81, 83, 85, 97, 103, 115, 121, 125, 127, 128, 131, 133, 147, 159, 171, 189, 227, 241, 243, 256, 269, 277, 289, 307, 311, 331, 335, 343, 361, 365, 367, 371, 391, 393, 399, 419, 425, 431, 439, 441, 477

Offset: 1

Emeric Deutsch, Oct 07 2011

nonn

The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.

The sequence is infinite.

			7 is in the sequence because the rooted tree with Matula-Goebel number 7 is the rooted tree Y, having all leaves at level 2.
2^m is in the sequence for each positive integer m because the rooted tree with Matula-Goebel number 2^m is a star with m edges.
From _Gus Wiseman_, Mar 30 2018: (Start)
Sequence of trees begins:
01 o
02 (o)
03 ((o))
04 (oo)
05 (((o)))
07 ((oo))
08 (ooo)
09 ((o)(o))
11 ((((o))))
16 (oooo)
17 (((oo)))
19 ((ooo))
21 ((o)(oo))
23 (((o)(o)))
25 (((o))((o)))
27 ((o)(o)(o))
31 (((((o)))))
(End)

F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.

Index entries for sequences related to Matula-Goebel numbers

Cf. A000081, A003238, A004111, A007097, A048816, A061775, A109082, A184154, A214577, A244925, A276625, A290689, A290760, A290822, A298422, A298424, A298426.

with(numtheory): P := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 1 elif bigomega(n) = 1 then sort(expand(x*P(pi(n)))) else sort(P(r(n))+P(s(n))) end if end proc: A := {}: for n to 500 do if degree(numer(subs(x = 1/x, P(n)))) = 0 then A := `union`(A, {n}) else  end if end do: A;

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
dep[n_]:=If[n===1,0,1+Max@@dep/@primeMS[n]];
rnkQ[n_]:=And[SameQ@@dep/@primeMS[n],And@@rnkQ/@primeMS[n]];
Select[Range[2000],rnkQ] (* Gus Wiseman, Mar 30 2018 *)

In A184154 one constructs for each n the generating polynomial P(n,x) of the leaves of the rooted tree with Matula-Goebel number n, according to their levels. The Maple program finds those n (between 1 and 500) for which P(n,x) is a monomial.

A324755 Number of integer partitions of n not containing 1 or any part whose prime indices all belong to the partition.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 4, 3, 5, 6, 10, 7, 16, 14, 23, 23, 35, 34, 53, 54, 75, 80, 112, 115, 160, 169, 223, 244, 315, 339, 442, 478, 604, 664, 832, 910, 1131, 1245, 1524, 1689, 2054, 2263, 2743, 3039, 3634, 4042, 4809, 5343, 6326, 7035, 8276, 9217, 10795, 12011

Offset: 0

Gus Wiseman, Mar 16 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

For example, (6,2) is such a partition because the prime indices of 6 are {1,2}, which do not all belong to the partition. On the other hand, (5,3) is not such a partition because the prime indices of 5 are {3}, and 3 belongs to the partition.

			The a(2) = 1 through a(10) = 10 integer partitions (A = 10):
  (2)  (3)  (4)   (5)  (6)    (7)   (8)     (9)    (A)
            (22)       (33)   (43)  (44)    (54)   (55)
                       (42)   (52)  (62)    (63)   (64)
                       (222)        (422)   (72)   (73)
                                    (2222)  (333)  (82)
                                            (522)  (433)
                                                   (442)
                                                   (622)
                                                   (4222)
                                                   (22222)

The subset version is A324739, with maximal case A324762. The strict case is A324750. The Heinz number version is A324760. An infinite version is A324694.

Cf. A000837, A001462, A051424, A112798, A276625, A290822, A304360, A306844, A324695, A324696, A324744.

Table[Length[Select[IntegerPartitions[n],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@If[k==1,{},FactorInteger[k]]]]&]],{n,0,30}]

A324759 Heinz numbers of integer partitions containing no part > 1 whose prime indices all belong to the partition.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 39, 40, 41, 43, 44, 46, 47, 49, 50, 51, 52, 53, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 71, 73, 74, 77, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93

Offset: 1

Gus Wiseman, Mar 17 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  10: {1,3}
  11: {5}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  23: {9}
  25: {3,3}
  26: {1,6}

The subset version is A324738, with maximal case A324744. The strict integer partition version is A324749. The integer partition version is A324754. An infinite version is A324694.

Cf. A000720, A001221, A007097, A056239, A112798, A276625, A289509, A290822, A306844, A324695, A324750, A324755, A324760.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!MemberQ[DeleteCases[primeMS[#],1],k_/;SubsetQ[primeMS[#],primeMS[k]]]&]

cp's OEIS Frontend

A324742 Number of subsets of {2...n} containing no prime indices of the elements.

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A352488 Weak nonexcedance set of A122111. Numbers k >= A122111(k), where A122111 represents partition conjugation using Heinz numbers.

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A324843 Number of unlabeled rooted trees with n nodes where the branches of any branch of any terminal subtree form a submultiset of the branches of the same subtree.

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A324935 Matula-Goebel numbers of rooted trees whose non-leaf terminal subtrees are all different.

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A301345 Regular triangle where T(n,k) is the number of transitive rooted trees with n nodes and k leaves.

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A324748 Number of strict integer partitions of n containing all prime indices of the parts.

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A358453 Number of transitive ordered rooted trees with n nodes.

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A184155 The Matula-Goebel number of rooted trees having all leaves at the same level.

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