A120383 -id:A120383 - cp's OEIS frontend

A324842 Matula-Goebel numbers of rooted trees where the branches of any branch of any terminal subtree form a submultiset of the branches of the same subtree.

1, 2, 4, 6, 8, 12, 16, 18, 24, 28, 32, 36, 48, 54, 56, 64, 72, 78, 84, 96, 108, 112, 128, 144, 152, 156, 162, 168, 192, 196, 216, 224, 234, 252, 256, 288, 304, 312, 324, 336, 384, 392, 432, 444, 448, 456, 468, 486, 504, 512, 576, 588, 608, 624, 648, 672, 702

Offset: 1

Gus Wiseman, Mar 18 2019

nonn

			The sequence of rooted trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   4: (oo)
   6: (o(o))
   8: (ooo)
  12: (oo(o))
  16: (oooo)
  18: (o(o)(o))
  24: (ooo(o))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  48: (oooo(o))
  54: (o(o)(o)(o))
  56: (ooo(oo))
  64: (oooooo)
  72: (ooo(o)(o))
  78: (o(o)(o(o)))
  84: (oo(o)(oo))
  96: (ooooo(o))

A subsequence of A120383.
Cf. A000081, A007097, A112798, A279861, A290689, A290760, A290822, A318186.
Cf. A324704, A324736, A324748, A324753, A324843, A324847, A324848, A324854.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
qaQ[n_]:=And[And@@Table[Divisible[n,x],{x,primeMS[n]}],And@@qaQ/@primeMS[n]];
Select[Range[1000],qaQ]

A387114 Number of divisors in common to all prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 1, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 1, 1, 6, 1, 2, 1, 2, 1, 4, 1, 1, 1, 4, 1, 3, 1, 1, 1, 5, 1, 1, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 2, 1, 1, 1, 6, 1, 4, 1, 1, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2

Offset: 1

Gus Wiseman, Aug 19 2025

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.

Also the number of divisors of the greatest common divisor of the prime indices of n.

			The prime indices of 703 are {8,12}, with divisors {{1,2,4,8},{1,2,3,4,6,12}}, with {1,2,4} in common, so a(703) = 3.

For initial interval instead of divisors we have A055396.
Positions of 1 are A289509, complement A318978.
Positions of 2 are A387119.
For prime factors or indices instead of divisors we have A387135, see A010055 or A069513.
A000005 counts divisors.
A001414 adds up distinct prime divisors, counted by A001221.
A003963 multiplies together the prime indices of n.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289508 gives greatest common divisor of prime indices.
Cf. A000720, A061395, A326841, A335433, A335448, A355731, A355733, A355734, A355737, A355739, A355740, A370820.

Table[If[n==1,0,Length[Divisors[GCD@@PrimePi/@First/@FactorInteger[n]]]],{n,100}]

a(1) = 0; a(n) = A000005(A289508(n)) for n > 1.

A387327 Number of ways to choose an integer partition of each prime factor of n (with multiplicity).

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 15, 8, 9, 14, 56, 12, 101, 30, 21, 16, 297, 18, 490, 28, 45, 112, 1255, 24, 49, 202, 27, 60, 4565, 42, 6842, 32, 168, 594, 105, 36, 21637, 980, 303, 56, 44583, 90, 63261, 224, 63, 2510, 124754, 48, 225, 98, 891, 404, 329931, 54, 392, 120

Offset: 1

Gus Wiseman, Sep 05 2025

			The a(1) = 1 through a(7) = 15 ways:
  (1)  (2)   (3)    (2)(2)    (5)      (2)(3)     (7)
       (11)  (21)   (11)(2)   (32)     (11)(3)    (43)
             (111)  (2)(11)   (41)     (2)(21)    (52)
                    (11)(11)  (221)    (11)(21)   (61)
                              (311)    (2)(111)   (322)
                              (2111)   (11)(111)  (331)
                              (11111)             (421)
                                                  (511)
                                                  (2221)
                                                  (3211)
                                                  (4111)
                                                  (22111)
                                                  (31111)
                                                  (211111)
                                                  (1111111)

For constant partitions we have A061142, for prime indices A355731.
For prime indices instead of factors we have A299200.
The version for distinct choices is A387133, zeros A387326.
A000041 counts integer partitions, strict A000009.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A387110 counts choices of distinct distinct integer partitions of each prime index.
Cf. A063834, A120383, A299201, A335433, A335448, A355741, A357977, A383706, A387115, A387120, A387136.

Table[Length[Tuples[IntegerPartitions/@Flatten[ConstantArray@@@FactorInteger[n]]]],{n,30}]

A387119 Numbers whose prime indices all have exactly 2 divisors in common.

Original entry on oeis.org

3, 5, 9, 11, 17, 21, 25, 27, 31, 39, 41, 57, 59, 63, 65, 67, 81, 83, 87, 91, 109, 111, 115, 117, 121, 125, 127, 129, 147, 157, 159, 171, 179, 183, 185, 189, 191, 203, 211, 213, 235, 237, 241, 243, 247, 261, 267, 273, 277, 283, 289, 299, 301, 303, 305, 319, 321

Offset: 1

Gus Wiseman, Aug 21 2025

nonn

All terms are odd.

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 prime indices of 87 are {2,10}, with divisors {{1,2},{1,2,5,10}}, with intersection {1,2}, so 87 is in the sequence.
The prime indices of 91 are {4,6}, with divisors {{1,2,4},{1,2,3,6}}, with intersection {1,2}, so 91 is in the sequence.
The terms together with their prime indices begin:
    3: {2}
    5: {3}
    9: {2,2}
   11: {5}
   17: {7}
   21: {2,4}
   25: {3,3}
   27: {2,2,2}
   31: {11}
   39: {2,6}
   41: {13}
   57: {2,8}
   59: {17}
   63: {2,2,4}
   65: {3,6}
   67: {19}
   81: {2,2,2,2}

For initial intervals instead of divisors we have A016945.
Positions of 1 are A289509, complement A318978.
Positions of 2 in A387114, for prime factors or indices A387135.
A000005 counts divisors.
A001414 adds up distinct prime divisors, counted by A001221.
A003963 multiplies together the prime indices of n.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289508 gives greatest common divisor of prime indices.
Cf. A000720, A055396, A061395, A335433, A355731, A355733, A355739, A355740, A370820.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],Length[Intersection@@Divisors/@prix[#]]==2&]

A387326 Numbers whose prime factors do not have choosable sets of integer partitions.

Original entry on oeis.org

8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 81

Offset: 1

Gus Wiseman, Sep 04 2025

We say that a set or sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1},{2},{1,3},{2,3}) is not.

Also numbers n with at least one prime index k such that the multiplicity of k in the prime factors of n exceeds the number of integer partitions of k.

			The prime factors of 72 are {2,2,2,3,3}, with sets of partitions ({(1,1),(2)},{(1,1),(2)},{(1,1),(2)},{(1,1),(2)},{(1,1,1),(2,1),(3)},{(1,1,1),(2,1),(3)}), which is not choosable, so 72 is in the sequence.

The version for prime indices differs from A276079 in lacking 16807, counted by A387134.
If we take the set {1..k} instead of the set of integer partitions of k we get A325127.
A subset of A365886.
Positions of zero in A387133.
For prime indices instead of factors we have A387577.
A000041 counts integer partitions, strict A000009.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A387327 counts partitions of prime factors.
A387328 counts partitions with choosable sets of partitions, ranks A387576.
Cf. A063834, A120383, A289509, A299200, A335433, A335448, A355739, A383706, A387111, A387135, A387180.

Select[Range[50],Select[Tuples[IntegerPartitions/@Join@@ConstantArray@@@FactorInteger[#]],UnsameQ@@#&]=={}&]

A387116 Number of ways to choose a constant sequence of integer partitions, one of each prime index of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 0, 5, 1, 2, 0, 7, 0, 11, 0, 0, 1, 15, 0, 22, 0, 0, 0, 30, 0, 3, 0, 2, 0, 42, 0, 56, 1, 0, 0, 0, 0, 77, 0, 0, 0, 101, 0, 135, 0, 0, 0, 176, 0, 5, 0, 0, 0, 231, 0, 0, 0, 0, 0, 297, 0, 385, 0, 0, 1, 0, 0, 490, 0, 0, 0, 627, 0, 792, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Aug 21 2025

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.

If n is a prime power prime(x)^y, then a(n) is the number of integer partitions of x; otherwise, a(n) = 0.

			The a(49) = 5 choices:
  ((4),(4))
  ((3,1),(3,1))
  ((2,2),(2,2))
  ((2,1,1),(2,1,1))
  ((1,1,1,1),(1,1,1,1))

Positions of zeros are A024619, complement A000961.
Twice-partitions of this type are counted by A047968, see also A296122.
For initial intervals instead of partitions we have A055396, see also A387111.
This is the constant case of A299200, see also A357977, A357982.
For disjoint instead of constant we have A383706.
For distinct instead of constant we have A387110.
For divisors instead of partitions we have A387114, see also A355731, A355739.
For strict partitions instead of partitions we have A387117.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together prime indices.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
Cf. A008284, A063834, A276078, A335433, A335448, A355529, A355741, A387115, A387120, A387133, A387136.

Table[Switch[n,1,1,?PrimePowerQ,PartitionsP[PrimePi[FactorInteger[n][[1,1]]]],,0],{n,100}]

a(n) = A000041(A297109(n)).

cp's OEIS Frontend

A324842 Matula-Goebel numbers of rooted trees where the branches of any branch of any terminal subtree form a submultiset of the branches of the same subtree.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A387114 Number of divisors in common to all prime indices of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A387327 Number of ways to choose an integer partition of each prime factor of n (with multiplicity).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A387119 Numbers whose prime indices all have exactly 2 divisors in common.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A387326 Numbers whose prime factors do not have choosable sets of integer partitions.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A387116 Number of ways to choose a constant sequence of integer partitions, one of each prime index of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula