A290822 -id:A290822 - cp's OEIS frontend

A318186 Totally transitive numbers. Matula-Goebel numbers of totally transitive rooted trees.

1, 2, 4, 6, 8, 12, 14, 16, 18, 24, 28, 32, 36, 38, 42, 48, 54, 56, 64, 72, 76, 78, 84, 96, 98, 106, 108, 112, 114, 126, 128, 144, 152, 156, 162, 168, 192, 196, 212, 216, 222, 224, 228, 234, 252, 256, 262, 266, 288, 294, 304, 312, 318, 324, 336, 342, 366, 378

Offset: 1

Gus Wiseman, Aug 20 2018

nonn

A number x is totally transitive if (1) whenever prime(y) divides x it follows that y is totally transitive and (2) if prime(y) divides x and prime(z) divides y then prime(z) also divides x.

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

Cf. A000081, A001678, A004111, A007097, A061775, A276625, A279861, A290689, A290760, A290822, A291636, A318185, A318187.

subprimes[n_]:=If[n==1,{},Union@@Cases[FactorInteger[n],{p_,_}:>FactorInteger[PrimePi[p]][[All,1]]]];
trmgQ[n_]:=Or[n==1,And[Divisible[n,Times@@subprimes[n]],And@@Cases[FactorInteger[n],{p_,_}:>trmgQ[PrimePi[p]]]]];
Select[Range[100],trmgQ]

A324737 Number of subsets of {2...n} containing every element of {2...n} whose prime indices all belong to the subset.

Original entry on oeis.org

1, 2, 3, 6, 8, 16, 24, 48, 84, 168, 216, 432, 648, 1296, 2448, 4896, 6528, 13056, 19584, 39168, 77760, 155520, 229248, 458496, 790272, 1580544, 3128832, 6257664, 9386496, 18772992, 24081408, 48162816, 95938560, 191877120, 378335232, 756670464, 1135005696, 2270011392

Offset: 1

Gus Wiseman, Mar 13 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.

Also the number of subsets of {2...n} with complement containing no term whose prime indices all belong to the subset.

			The a(1) = 1 through a(6) = 16 subsets:
  {}  {}   {}     {}       {}         {}
      {2}  {3}    {3}      {4}        {4}
           {2,3}  {4}      {5}        {5}
                  {2,3}    {3,5}      {6}
                  {3,4}    {4,5}      {3,5}
                  {2,3,4}  {2,3,5}    {4,5}
                           {3,4,5}    {4,6}
                           {2,3,4,5}  {5,6}
                                      {2,3,5}
                                      {3,4,5}
                                      {3,5,6}
                                      {4,5,6}
                                      {2,3,4,5}
                                      {2,3,5,6}
                                      {3,4,5,6}
                                      {2,3,4,5,6}
An example for n = 15 is {2, 3, 5, 8, 9, 10, 11, 15}. The numbers from 2 to 15 with all prime indices in the subset are {3, 5, 9, 11, 15}, which all belong to the subset, as required.

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

Cf. A000720, A001221, A001462, A007097, A084422, A085945, A112798, A276625, A290689, A290822, A304360, A306844.

Cf. A324697, A324698, A324736, A324738, A324748, A324753, A324755.

Table[Length[Select[Subsets[Range[2,n]],Function[set,SubsetQ[set,Select[Range[2,n],SubsetQ[set,PrimePi/@First/@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+(1<Andrew Howroyd, Aug 24 2019

Terms a(21) and beyond from Andrew Howroyd, Aug 24 2019

A324750 Number of strict 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, 1, 1, 2, 3, 2, 4, 4, 4, 6, 8, 8, 11, 10, 15, 16, 19, 23, 27, 28, 35, 39, 47, 50, 63, 68, 77, 91, 102, 114, 130, 147, 169, 187, 213, 237, 268, 300, 336, 380, 422, 472, 525, 587, 647, 731, 810, 895, 996, 1102, 1227, 1355, 1498, 1661, 1818, 2020, 2221

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 a(2) = 1 through a(17) = 15 strict integer partitions (A...H = 10...17):
  2  3  4  5  6   7   8   9   A   B    C    D    E    F    G    H
              42  43  62  54  64  65   75   76   86   87   97   98
                  52      63  73  83   84   85   95   96   A6   A7
                          72  82  542  93   94   A4   A5   C4   B6
                                       A2   A3   B3   B4   D3   C5
                                       642  B2   C2   C3   E2   D4
                                            643  752  D2   763  E3
                                            652  842  654  862  F2
                                                      762  943  854
                                                      843  A42  863
                                                      852       872
                                                                A43
                                                                A52
                                                                B42
                                                                6542

The subset version is A324739. The non-strict version is A324755. The Heinz number version is A324760. An infinite version is A324694.
Cf. A000720, A001462, A007097, A074971, A078374, A112798, A276625, A290822, A304360, A305713, A306844.
Cf. A324696, A324737, A324742, A324744, A324764.

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

A324754 Number of integer partitions of n containing no part > 1 whose prime indices all belong to the partition.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 7, 8, 11, 12, 19, 19, 30, 34, 46, 50, 71, 76, 104, 119, 151, 171, 225, 247, 315, 360, 446, 504, 629, 703, 867, 986, 1192, 1346, 1636, 1837, 2204, 2500, 2965, 3348, 3980, 4475, 5276, 5963, 6973, 7852, 9194, 10335, 12009, 13536, 15650, 17589

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(1) = 1 through a(8) = 11  integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (43)       (44)
                    (31)    (11111)  (42)      (52)       (62)
                    (1111)           (51)      (61)       (71)
                                     (222)     (331)      (422)
                                     (3111)    (511)      (611)
                                     (111111)  (31111)    (2222)
                                               (1111111)  (3311)
                                                          (5111)
                                                          (311111)
                                                          (11111111)

The subset version is A324738, with maximal case A324744. The strict case is A324749. The Heinz number version is A324759. An infinite version is A324694.

Cf. A000837, A001462, A007097, A051424, A112798, A276625, A290822, A304360, A306844, A324695, A324750, A324755, A324760.

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

A324760 Heinz numbers of integer partitions not containing 1 or any part whose prime indices all belong to the partition.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 65, 67, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 117, 121, 123, 125, 127, 129, 131, 133, 137, 139

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: {}
   3: {2}
   5: {3}
   7: {4}
   9: {2,2}
  11: {5}
  13: {6}
  17: {7}
  19: {8}
  21: {2,4}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  33: {2,5}
  35: {3,4}
  37: {12}
  39: {2,6}
  41: {13}

The subset version is A324739, with maximal case A324762. The strict integer partition version is A324750. The integer partition version is A324755. An infinite version is A324694.

Cf. A000720, A001221, A007097, A056239, A112798, A289509, A290822, A306844, A324695, A324696, A324737, A324744.

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

A325756 A number k belongs to the sequence if k = 1 or k is divisible by its prime shadow A181819(k) and the quotient k/A181819(k) also belongs to the sequence.

Original entry on oeis.org

1, 2, 12, 336, 360, 45696, 52416, 75600, 22665216, 31804416, 42928704, 77792400, 92610000, 164656800, 174636000

Offset: 1

Gus Wiseman, May 19 2019

We define the prime shadow A181819(k) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.

			The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
     12: {1,1,2}
    336: {1,1,1,1,2,4}
    360: {1,1,1,2,2,3}
  45696: {1,1,1,1,1,1,1,2,4,7}
  52416: {1,1,1,1,1,1,2,2,4,6}
  75600: {1,1,1,1,2,2,2,3,3,4}

Cf. A181819, A182857, A290689, A290822, A323014, A324843, A325702, A325706, A325708, A325755.

red[n_] := If[n == 1, 1, Times @@ Prime /@ Last /@ FactorInteger[n]];
suQ[n_]:=n==1||Divisible[n,red[n]]&&suQ[n/red[n]];
Select[Range[10000],suQ]

ps(n) = my(f=factor(n)); prod(k=1, #f~, prime(f[k, 2])); \\ A181819
isok(k) = {if ((k==1), return(1)); my(p=ps(k)); ((k % p) == 0) && isok(k/p);} \\ Michel Marcus, Jan 09 2021

a(9)-a(15) from Amiram Eldar, Jan 09 2021

A318229 Number of inequivalent leaf-colorings of transitive rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 5, 13, 34, 92, 255

Offset: 1

Gus Wiseman, Aug 21 2018

In a transitive rooted tree, every branch of a branch of the root is also a branch of the root.

			Inequivalent representatives of the a(5) = 13 leaf-colorings:
  (1111)  (1(11))  (11(1))
  (1112)  (1(12))  (11(2))
  (1122)  (1(22))  (12(1))
  (1123)  (1(23))  (12(3))
  (1234)

Cf. A000081, A001190, A001678, A003238, A004111, A279861, A290689, A290822, A318185, A304486.

Cf. A318226, A318227, A318228, A318230, A318231, A318234.

A324739 Number of subsets of {2...n} containing no element whose prime indices all belong to the subset.

Original entry on oeis.org

1, 2, 3, 6, 10, 20, 30, 60, 96, 192, 312, 624, 936, 1872, 3744, 7488, 12480, 24960, 37440, 74880, 142848, 285696, 456192, 912384, 1548288, 3096576, 5308416, 10616832, 15925248, 31850496, 51978240, 103956480, 200835072, 401670144, 771489792, 1542979584, 2314469376

Offset: 1

Gus Wiseman, Mar 14 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) = 20 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}    {2,6}
                       {2,4,5}  {3,4}
                                {3,6}
                                {4,5}
                                {4,6}
                                {5,6}
                                {2,4,5}
                                {2,4,6}
                                {2,5,6}
                                {3,4,6}
                                {4,5,6}
                                {2,4,5,6}

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

The maximal case is A324762. The case of subsets of {1...n} is A324738. The strict integer partition version is A324750. The integer partition version is A324755. The Heinz number version is A324760. An infinite version is A324694.
Cf. A000720, A001221, A001462, A007097, A084422, A085945, A112798, A276625, A279861, A290689, A290822, A304360, A306844.
Cf. A324695, A324696, A324736, A324737, A324741, A324742, A324744, A324764.

Table[Length[Select[Subsets[Range[2,n]],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]],{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,k,pset(k)), d=0); for(i=1, #p, d=bitor(d, p[i]));
((k,b)->if(k>#p, 1, my(t=self()(k+1,b)); if(bitnegimply(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

A324749 Number of strict integer partitions of n containing no part > 1 whose prime indices all belong to the partition.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 4, 3, 4, 6, 6, 8, 11, 10, 14, 14, 19, 21, 26, 28, 35, 38, 44, 50, 60, 65, 79, 88, 98, 113, 131, 144, 165, 185, 211, 234, 268, 297, 334, 374, 420, 470, 525, 584, 649, 727, 801, 902, 998, 1100, 1220, 1357, 1500, 1657, 1833, 2029, 2220, 2462

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 a(0) = 1 through a(10) = 6 strict integer partitions:
  ()  (1)  (2)  (3)  (4)    (5)  (6)    (7)    (8)    (9)    (10)
                     (3,1)       (4,2)  (4,3)  (6,2)  (5,4)  (6,4)
                                 (5,1)  (5,2)  (7,1)  (6,3)  (7,3)
                                        (6,1)         (7,2)  (8,2)
                                                             (9,1)
                                                             (6,3,1)

The subset version is A324738. The non-strict version is A324754. The Heinz number version is A324759. An infinite version is A324694.
Cf. A000720, A001462, A007097, A074971, A078374, A112798, A276625, A290822, A305713.
Cf. A324744, A324750, A324755, A324760.

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

A324771 Numbers divisible by at least one of their prime indices > 1.

Original entry on oeis.org

6, 12, 15, 18, 24, 28, 30, 36, 42, 45, 48, 54, 55, 56, 60, 66, 72, 75, 78, 84, 90, 96, 102, 105, 108, 110, 112, 114, 119, 120, 126, 132, 135, 138, 140, 144, 150, 152, 156, 162, 165, 168, 174, 180, 186, 192, 195, 196, 198, 204, 207, 210, 216, 220, 222, 224, 225

Offset: 1

Gus Wiseman, Mar 18 2019

nonn

A prime index of n is a number m such that prime(m) divides n.

			The sequence of terms together with their prime indices begins:
   6: {1,2}
  12: {1,1,2}
  15: {2,3}
  18: {1,2,2}
  24: {1,1,1,2}
  28: {1,1,4}
  30: {1,2,3}
  36: {1,1,2,2}
  42: {1,2,4}
  45: {2,2,3}
  48: {1,1,1,1,2}
  54: {1,2,2,2}
  55: {3,5}
  56: {1,1,1,4}
  60: {1,1,2,3}
  66: {1,2,5}
  72: {1,1,1,2,2}
  75: {2,3,3}
  78: {1,2,6}
  84: {1,1,2,4}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement of A324849.
Cf. A003963, A056239, A112798, A120383, A289509, A290822, A304360.
Cf. A324695, A324846, A324847, A324848, A324850, A324851, A324852, A324853.

Select[Range[100],Or@@Cases[If[#==1,{},FactorInteger[#]],{p_?(#>2&),_}:>Divisible[#,PrimePi[p]]]&]

cp's OEIS Frontend

A318186 Totally transitive numbers. Matula-Goebel numbers of totally transitive rooted trees.

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A324737 Number of subsets of {2...n} containing every element of {2...n} whose prime indices all belong to the subset.

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A324750 Number of strict integer partitions of n not containing 1 or any part whose prime indices all belong to the partition.

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A324754 Number of integer partitions of n containing no part > 1 whose prime indices all belong to the partition.

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A324760 Heinz numbers of integer partitions not containing 1 or any part whose prime indices all belong to the partition.

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A325756 A number k belongs to the sequence if k = 1 or k is divisible by its prime shadow A181819(k) and the quotient k/A181819(k) also belongs to the sequence.

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A318229 Number of inequivalent leaf-colorings of transitive rooted trees with n nodes.

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A324739 Number of subsets of {2...n} containing no element whose prime indices all belong to the subset.

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A324749 Number of strict integer partitions of n containing no part > 1 whose prime indices all belong to the partition.