A302796 -id:A302796 - cp's OEIS frontend

A316502 Matula-Goebel numbers of unlabeled rooted trees with n nodes in which the branches of any node with more than one branch have empty intersection.

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 47, 48, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 70, 71, 72, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Gus Wiseman, Jul 05 2018

nonn

A prime index of n is a number m such that prime(m) divides n. A number is in the sequence iff it is 1, or either it is a prime or its prime indices are relatively prime, and its prime indices already belong to the sequence.

			Sequence of rooted trees preceded by 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))
  15: ((o)((o)))
  16: (oooo)

Index entries for sequences related to Matula-Göbel numbers

Cf. A000081, A000837, A007097, A276625, A289509, A302796, A316468, A316470, A316495, A316501, A316502.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
go[n_]:=Or[n==1,If[PrimeQ[n],go[PrimePi[n]],And[GCD@@primeMS[n]==1,And@@go/@primeMS[n]]]]
Select[Range[100],go]

A302798 Squarefree numbers that are prime or whose prime indices are pairwise coprime. Heinz numbers of strict integer partitions that either consist of a single part or have pairwise coprime parts.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 41, 43, 46, 47, 51, 53, 55, 58, 59, 61, 62, 66, 67, 69, 70, 71, 73, 74, 77, 79, 82, 83, 85, 86, 89, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109, 110, 113, 118, 119, 122

Offset: 1

Gus Wiseman, Apr 13 2018

nonn

A prime index of n is a number m such that prime(m) divides n. Two or more numbers are coprime if no pair of them has a common divisor other than 1. A single number is not considered coprime unless it is equal to 1.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			Sequence of terms together with their sets of prime indices begins:
01 : {}
02 : {1}
03 : {2}
05 : {3}
06 : {1,2}
07 : {4}
10 : {1,3}
11 : {5}
13 : {6}
14 : {1,4}
15 : {2,3}
17 : {7}
19 : {8}
22 : {1,5}
23 : {9}
26 : {1,6}
29 : {10}
30 : {1,2,3}

Cf. A001222, A003963, A005117, A007359, A051424, A056239, A275024, A289509, A294472, A302242, A302505, A302696, A302697, A302698, A302796, A302797.

Select[Range[100],Or[#===1,SquareFreeQ[#]&&(PrimeQ[#]||CoprimeQ@@PrimePi/@FactorInteger[#][[All,1]])]&]

A303138 Regular triangle where T(n,k) is the number of strict integer partitions of n with greatest common divisor k.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 4, 0, 0, 0, 0, 0, 1, 4, 1, 0, 0, 0, 0, 0, 1, 6, 0, 1, 0, 0, 0, 0, 0, 1, 7, 2, 0, 0, 0, 0, 0, 0, 0, 1, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 10, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 17, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 23, 0, 2, 0, 1

Offset: 1

Gus Wiseman, Apr 19 2018

			Triangle begins:
01:   1
02:   0  1
03:   1  0  1
04:   1  0  0  1
05:   2  0  0  0  1
06:   2  1  0  0  0  1
07:   4  0  0  0  0  0  1
08:   4  1  0  0  0  0  0  1
09:   6  0  1  0  0  0  0  0  1
10:   7  2  0  0  0  0  0  0  0  1
11:  11  0  0  0  0  0  0  0  0  0  1
12:  10  2  1  1  0  0  0  0  0  0  0  1
13:  17  0  0  0  0  0  0  0  0  0  0  0  1
14:  17  4  0  0  0  0  0  0  0  0  0  0  0  1
15:  23  0  2  0  1  0  0  0  0  0  0  0  0  0  1
The strict partitions counted in row 12 are the following.
T(12,1) = 10: (11,1) (9,2,1) (8,3,1) (7,5) (7,4,1) (7,3,2) (6,5,1) (6,3,2,1) (5,4,3) (5,4,2,1)
T(12,2) = 2:  (10,2) (6,4,2)
T(12,3) = 1:  (9,3)
T(12,4) = 1:  (8,4)
T(12,12) = 1: (12)

First column is A078374. Second column at even indices is same as first column. Row sums are A000009. Row sums with first column removed are A303280.

Cf. A000009, A000837, A018783, A051424, A117408, A168532, A289508, A289509, A298748, A300486, A302698, A302796, A303139, A303140, A303280.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&GCD@@#===k&]],{n,15},{k,n}]

If k divides n, T(n,k) = A078374(n/k); otherwise T(n,k) = 0.

A316501 Number of unlabeled rooted trees with n nodes in which the branches of any node with more than one distinct branch have empty intersection.

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 45, 103, 250, 611, 1528, 3853, 9875, 25481, 66382, 174085, 459541, 1219462

Offset: 1

Gus Wiseman, Jul 05 2018

			The a(6) = 19 rooted trees:
  (((((o)))))
  ((((oo))))
  (((o(o))))
  (((ooo)))
  ((o((o))))
  ((o(oo)))
  (((o)(o)))
  ((oo(o)))
  ((oooo))
  (o(((o))))
  (o((oo)))
  (o(o(o)))
  (o(ooo))
  ((o)((o)))
  (oo((o)))
  (oo(oo))
  (o(o)(o))
  (ooo(o))
  (ooooo)

Cf. A000081, A007562, A289509, A302796, A316470, A316473, A316475, A316500, A316502.

strut[n_]:=strut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[strut/@c]]]/@IntegerPartitions[n-1],Or[Length[Union[#]]==1,Intersection@@#=={}]&]];
Table[Length[strut[n]],{n,15}]

A328868 Heinz numbers of integer partitions with no two (not necessarily distinct) parts relatively prime, but with no divisor in common to all of the parts.

Original entry on oeis.org

17719, 40807, 43381, 50431, 74269, 83143, 101543, 105703, 116143, 121307, 123469, 139919, 140699, 142883, 171613, 181831, 185803, 191479, 203557, 205813, 211381, 213239, 215267, 219271, 230347, 246703, 249587, 249899, 279371, 286897, 289007, 296993, 300847

Offset: 1

Gus Wiseman, Oct 30 2019

nonn

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:
   17719: {6,10,15}
   40807: {6,14,21}
   43381: {6,15,20}
   50431: {10,12,15}
   74269: {6,10,45}
   83143: {10,15,18}
  101543: {6,21,28}
  105703: {6,15,40}
  116143: {12,14,21}
  121307: {10,15,24}
  123469: {12,15,20}
  139919: {6,15,50}
  140699: {6,22,33}
  142883: {6,10,75}
  171613: {6,14,63}
  181831: {6,20,45}
  185803: {10,14,35}
  191479: {14,18,21}
  203557: {15,18,20}
  205813: {10,15,36}
  211381: {10,12,45}
  213239: {6,15,70}
  215267: {6,10,105}
  219271: {6,26,39}
  230347: {6,6,10,15}

These are the Heinz numbers of the partitions counted by A202425.
Terms of A328679 that are not powers of 2.
The strict case is A318716 (preceded by 2).
A ranking using binary indices (instead of prime indices) is A326912.
Heinz numbers of relatively prime partitions are A289509.
Cf. A000837, A056239, A112798, A200976, A291166, A302796, A316476, A318715, A319752, A319759, A328336, A328672, A328677, A328867.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
dv=Select[Range[100000],GCD@@primeMS[#]==1&&And[And@@(GCD[##]>1&)@@@Tuples[Union[primeMS[#]],2]]&]

A303365 Number of integer partitions of the n-th squarefree number using squarefree numbers.

Original entry on oeis.org

1, 2, 3, 6, 9, 12, 28, 36, 60, 76, 96, 150, 228, 342, 416, 504, 877, 1484, 1759, 2079, 2885, 3387, 3968, 5413, 6304, 7328, 9852, 11395, 13159, 20082, 23056, 39532, 51385, 66488, 85660, 97078, 109907, 140465, 158573, 226918, 255268, 286920, 361606, 405470

Offset: 1

Gus Wiseman, Apr 22 2018

nonn

			The a(5) = 9 partitions are (6), (51), (33), (321), (3111), (222), (2211), (21111), (111111).

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000051, A005117, A072774, A073576, A078374, A302491, A302796, A303283, A303364.

nn=80;
sqf=Select[Range[nn],SquareFreeQ];
ser=Product[1/(1-x^sqf[[n]]),{n,Length[sqf]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,sqf}]

a(n) = A073576(A005117(n)).

A316469 Matula-Goebel numbers of unlabeled rooted identity RPMG-trees, meaning the Matula-Goebel numbers of the branches of any non-leaf node are relatively prime.

Original entry on oeis.org

1, 2, 6, 26, 78, 202, 606, 794, 2382, 2462, 2626, 7386, 7878, 8914, 10322, 12178, 26742, 30966, 32006, 36534, 42374, 43954, 47206, 80194, 96018, 115882, 127122, 131862, 141618, 149782, 158314, 160978, 184622, 217058, 240582, 248662, 260422, 347646, 449346

Offset: 1

Gus Wiseman, Jul 04 2018

nonn

A prime index of n is a number m such that prime(m) divides n. A number is in the sequence iff it is 1 or its prime indices are distinct, relatively prime, and already belong to the sequence.

			78 = prime(1)*prime(2)*prime(6) belongs to the sequence because the indices {1,2,6} are relatively prime, distinct, and already belong to the sequence.
The sequence of all identity RPMG-trees preceded by their Matula-Goebel numbers begins:
     1: o
     2: (o)
     6: (o(o))
    26: (o(o(o)))
    78: (o(o)(o(o)))
   202: (o(o(o(o))))
   606: (o(o)(o(o(o))))
   794: (o(o(o)(o(o))))
  2382: (o(o)(o(o)(o(o))))
  2462: (o(o(o(o(o)))))
  2626: (o(o(o))(o(o(o))))
  7386: (o(o)(o(o(o(o)))))
  7878: (o(o)(o(o))(o(o(o))))

Cf. A000081, A000837, A004111, A007097, A078374, A276625, A289509, A302696, A302796, A316467, A316470, A316471, A316474, A316494.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Or[#==1,And[SquareFreeQ[#],GCD@@primeMS[#]==1,And@@#0/@primeMS[#]]]&]

A316503 Matula-Goebel numbers of unlabeled rooted identity trees with n nodes in which the branches of any node with more than one branch have empty intersection.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 13, 15, 22, 26, 29, 30, 31, 33, 41, 47, 55, 58, 62, 66, 78, 79, 82, 93, 94, 101, 109, 110, 113, 123, 127, 130, 137, 141, 143, 145, 155, 158, 165, 174, 179, 186, 195, 202, 205, 211, 218, 226, 246, 254, 257, 271, 274, 282, 286, 290, 293

Offset: 1

Gus Wiseman, Jul 05 2018

nonn

			Sequence of rooted identity trees preceded by their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   5: (((o)))
   6: (o(o))
  10: (o((o)))
  11: ((((o))))
  13: ((o(o)))
  15: ((o)((o)))
  22: (o(((o))))
  26: (o(o(o)))
  29: ((o((o))))
  30: (o(o)((o)))
  31: (((((o)))))

Cf. A000081, A000837, A004111, A007097, A078374, A276625, A289509, A302796, A316467, A316470, A316494, A316502.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Or[#==1,And[SquareFreeQ[#],Or[PrimeQ[#],GCD@@primeMS[#]==1],And@@#0/@primeMS[#]]]&]

A355738 Least k such that there are exactly n ways to choose a sequence of divisors, one of each prime index of k (with multiplicity), such that the result has no common divisor > 1.

Original entry on oeis.org

1, 2, 6, 9, 15, 49, 35, 27, 45, 98, 63, 105, 171, 117, 81, 135, 245, 343, 273, 549, 189, 1083, 315, 5618, 741, 686, 507, 513, 351, 243, 405, 7467, 6419, 5575, 735, 6859, 1813, 3231, 1183, 1197, 3537, 819, 1647, 567, 945, 2197, 8397, 3211, 1715, 3249, 3367

Offset: 1

Gus Wiseman, Jul 21 2022

nonn

This is the position of first appearance of n in A355737.

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 terms together with their prime indices begin:
     1: {}
     2: {1}
     6: {1,2}
     9: {2,2}
    15: {2,3}
    49: {4,4}
    35: {3,4}
    27: {2,2,2}
    45: {2,2,3}
    98: {1,4,4}
    63: {2,2,4}
   105: {2,3,4}
   171: {2,2,8}
   117: {2,2,6}
    81: {2,2,2,2}
   135: {2,2,2,3}
For example, the choices for a(12) = 105 are:
  (1,1,1)  (1,3,2)  (2,1,4)
  (1,1,2)  (1,3,4)  (2,3,1)
  (1,1,4)  (2,1,1)  (2,3,2)
  (1,3,1)  (2,1,2)  (2,3,4)

Wikipedia, Coprime integers.

Not requiring coprimality gives A355732, firsts of A355731.
Positions of first appearances in A355737.
A000005 counts divisors.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A120383 lists numbers divisible by all of their prime indices.
A289508 gives GCD of prime indices.
A289509 ranks relatively prime partitions, odd A302697, squarefree A302796.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000720, A007359, A051424, A076610, A302696, A302698, A355733, A355735, A355739, A355741, A355748.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
az=Table[Length[Select[Tuples[Divisors/@primeMS[n]],GCD@@#==1&]],{n,100}];
Table[Position[az+1,k][[1,1]],{k,mnrm[az+1]}]

A328679 Heinz numbers of integer partitions with no two distinct parts relatively prime, but with no divisor in common to all of the parts.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 17719, 32768, 40807, 43381, 50431, 65536, 74269, 83143, 101543, 105703, 116143, 121307, 123469, 131072, 139919, 140699, 142883, 171613, 181831, 185803, 191479, 203557, 205813, 211381, 213239

Offset: 1

Gus Wiseman, Oct 30 2019

nonn

Equals the union A000079 and A328868.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
A partition with no two distinct parts relatively prime is said to be intersecting.

			The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
      4: {1,1}
      8: {1,1,1}
     16: {1,1,1,1}
     32: {1,1,1,1,1}
     64: {1,1,1,1,1,1}
    128: {1,1,1,1,1,1,1}
    256: {1,1,1,1,1,1,1,1}
    512: {1,1,1,1,1,1,1,1,1}
   1024: {1,1,1,1,1,1,1,1,1,1}
   2048: {1,1,1,1,1,1,1,1,1,1,1}
   4096: {1,1,1,1,1,1,1,1,1,1,1,1}
   8192: {1,1,1,1,1,1,1,1,1,1,1,1,1}
  16384: {1,1,1,1,1,1,1,1,1,1,1,1,1,1}
  17719: {6,10,15}
  32768: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
  40807: {6,14,21}
  43381: {6,15,20}
  50431: {10,12,15}
  65536: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}

These are the Heinz numbers of the partitions counted by A328672.
Terms that are not powers of 2 are A328868.
The strict case is A318716.
The version without global relative primality is A328867.
A ranking using binary indices (instead of prime indices) is A326912.
The version for non-isomorphic multiset partitions is A319759.
The version for divisibility (instead of relative primality) is A328677.
Heinz numbers of relatively prime partitions are A289509.
Cf. A000837, A056239, A112798, A200976, A202425, A289509, A291166, A302796, A316476, A318715, A319752, A328336.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[10000],#==1||GCD@@primeMS[#]==1&&And[And@@(GCD[##]>1&)@@@Subsets[Union[primeMS[#]],{2}]]&]

cp's OEIS Frontend

A316502 Matula-Goebel numbers of unlabeled rooted trees with n nodes in which the branches of any node with more than one branch have empty intersection.

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A302798 Squarefree numbers that are prime or whose prime indices are pairwise coprime. Heinz numbers of strict integer partitions that either consist of a single part or have pairwise coprime parts.

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A303138 Regular triangle where T(n,k) is the number of strict integer partitions of n with greatest common divisor k.

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A316501 Number of unlabeled rooted trees with n nodes in which the branches of any node with more than one distinct branch have empty intersection.

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A328868 Heinz numbers of integer partitions with no two (not necessarily distinct) parts relatively prime, but with no divisor in common to all of the parts.

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A303365 Number of integer partitions of the n-th squarefree number using squarefree numbers.

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A316469 Matula-Goebel numbers of unlabeled rooted identity RPMG-trees, meaning the Matula-Goebel numbers of the branches of any non-leaf node are relatively prime.

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A316503 Matula-Goebel numbers of unlabeled rooted identity trees with n nodes in which the branches of any node with more than one branch have empty intersection.

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A355738 Least k such that there are exactly n ways to choose a sequence of divisors, one of each prime index of k (with multiplicity), such that the result has no common divisor > 1.

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