A285573 -id:A285573 - cp's OEIS frontend

A318746 Number of Lyndon compositions (aperiodic necklaces of positive integers) with sum n and successive parts (including the last with the first part) being indivisible.

1, 1, 1, 1, 2, 1, 3, 2, 4, 5, 6, 8, 11, 17, 20, 29, 41, 56, 79, 107, 155, 214, 305, 422, 604, 850, 1207, 1709, 2424, 3439, 4905, 6972, 9949, 14171, 20268, 28915, 41392, 59176, 84790, 121428, 174163, 249760, 358578, 514873, 739910, 1063523, 1529767, 2200926

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

			The a(14) = 17 Lyndon compositions with successive parts indivisible:
  (14)
  (3,11) (4,10) (5,9) (6,8)
  (2,3,9) (2,5,7) (2,7,5) (3,4,7) (3,6,5) (3,7,4)
  (2,3,2,7) (2,3,4,5) (2,4,3,5) (2,4,5,3) (2,5,4,3)
  (2,3,2,4,3)

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

Cf. A000740, A008965, A059966, A285573, A303362, A318726, A318727, A318729, A318730, A318731, A318745, A318747.

LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[Length[#]==1,LyndonQ[#]&&And@@Not/@Divisible@@@Partition[#,2,1,1]]&]],{n,20}]

b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q, ]}
seq(n)={my(v=sum(k=1, n, k*b(n, k, (i, j)->i%j<>0))); vector(n, n, 1 + sumdiv(n, d, moebius(d)*v[n/d])/n)} \\ Andrew Howroyd, Nov 01 2019

Terms a(21) and beyond from Andrew Howroyd, Sep 08 2018

A318747 Number of Lyndon compositions (aperiodic necklaces of positive integers) with sum n and adjacent parts (including the last with the first part) being indivisible (either way).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 3, 5, 5, 8, 7, 12, 14, 20, 31, 37, 51, 64, 96, 129, 177, 246, 328, 465, 630, 889, 1230, 1692, 2370, 3250, 4587, 6354, 8895, 12384, 17252, 24180, 33777, 47336, 66254, 92752, 130142, 182337, 256246, 359500, 505231, 709787, 997951, 1403883

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

			The a(14) = 12 Lyndon compositions with adjacent parts indivisible either way:
  (14)
  (3,11) (4,10) (5,9) (6,8)
  (2,5,7) (2,7,5) (3,4,7) (3,7,4)
  (2,3,2,7) (2,3,4,5) (2,5,4,3)

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

Cf. A000740, A008965, A059966, A285573, A303362, A318726, A318727, A318729, A318730, A318731, A318745, A318746.

LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[Length[#]==1,And[LyndonQ[#],And@@Not/@Divisible@@@Partition[#,2,1,1],And@@Not/@Divisible@@@Reverse/@Partition[#,2,1,1]]]&]],{n,20}]

b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q, ]}
seq(n)={my(v=sum(k=1, n, k*b(n, k, (i, j)->i%j<>0 && j%i<>0))); vector(n, n, 1 + sumdiv(n, d, moebius(d)*v[n/d])/n)} \\ Andrew Howroyd, Nov 01 2019

Terms a(21) and beyond from Andrew Howroyd, Sep 08 2018

A319496 Numbers whose prime indices are distinct and pairwise indivisible and whose own prime indices are connected and span an initial interval of positive integers.

Original entry on oeis.org

2, 3, 7, 13, 19, 37, 53, 61, 89, 91, 113, 131, 151, 223, 247, 251, 281, 299, 311, 359, 377, 427, 463, 503, 593, 611, 659, 689, 703, 719, 791, 827, 851, 863, 923, 953, 1069, 1073, 1159, 1163, 1291, 1321, 1339, 1363, 1511, 1619, 1703, 1733, 1739, 1757, 1769

Offset: 1

Gus Wiseman, Dec 16 2018

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 multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}. This sequence lists all MM-numbers of connected strict antichains of multisets spanning an initial interval of positive integers.

			The sequence of multisystems whose MM-numbers belong to the sequence begins:
    2: {{}}
    3: {{1}}
    7: {{1,1}}
   13: {{1,2}}
   19: {{1,1,1}}
   37: {{1,1,2}}
   53: {{1,1,1,1}}
   61: {{1,2,2}}
   89: {{1,1,1,2}}
   91: {{1,1},{1,2}}
  113: {{1,2,3}}
  131: {{1,1,1,1,1}}
  151: {{1,1,2,2}}
  223: {{1,1,1,1,2}}
  247: {{1,2},{1,1,1}}
  251: {{1,2,2,2}}
  281: {{1,1,2,3}}
  299: {{1,2},{2,2}}

Cf. A003963, A006126, A055932, A056239, A112798, A285573, A286520, A293994, A302242, A318401, A319719, A319837, A320275, A320456, A320532.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[200],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],stableQ[primeMS[#],Divisible],Length[zsm[primeMS[#]]]==1]&]

A371179 Positive integers with fewer distinct prime factors (A001221) than distinct divisors of prime indices (A370820).

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 14, 15, 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 33, 35, 37, 38, 39, 41, 43, 45, 46, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 63, 65, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 98, 99, 101

Offset: 1

Gus Wiseman, Mar 19 2024

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 terms together with their prime indices begin:
     3: {2}        28: {1,1,4}    52: {1,1,6}      74: {1,12}
     5: {3}        29: {10}       53: {16}         75: {2,3,3}
     7: {4}        31: {11}       55: {3,5}        76: {1,1,8}
     9: {2,2}      33: {2,5}      56: {1,1,1,4}    77: {4,5}
    11: {5}        35: {3,4}      57: {2,8}        78: {1,2,6}
    13: {6}        37: {12}       58: {1,10}       79: {22}
    14: {1,4}      38: {1,8}      59: {17}         81: {2,2,2,2}
    15: {2,3}      39: {2,6}      61: {18}         83: {23}
    17: {7}        41: {13}       63: {2,2,4}      85: {3,7}
    19: {8}        43: {14}       65: {3,6}        86: {1,14}
    21: {2,4}      45: {2,2,3}    67: {19}         87: {2,10}
    23: {9}        46: {1,9}      69: {2,9}        89: {24}
    25: {3,3}      47: {15}       70: {1,3,4}      91: {4,6}
    26: {1,6}      49: {4,4}      71: {20}         92: {1,1,9}
    27: {2,2,2}    51: {2,7}      73: {21}         93: {2,11}

The LHS is A001221, distinct case of A001222.
The RHS is A370820, for prime factors A303975.
Partitions of this type are counted by A371132, strict A371180.
Counting all prime indices on the LHS gives A371168, counted by A371173.
The complement is A371177, counted by A371178, strict A371128.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
A305148 counts pairwise indivisible (stable) partitions, ranks A316476.
Cf. A003963, A239312, A285573, A355529, A370802, A370803, A371130, A371165, A371171, A371172.

Mathematica
```
Select[Range[100],PrimeNu[#]
				
```

A001221(a(n)) < A370820(a(n)).

A317101 Numbers whose prime multiplicities are pairwise indivisible.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87

Offset: 1

Gus Wiseman, Aug 01 2018

nonn

			72 = 2^3 * 3^2 is in the sequence because 3 and 2 are pairwise indivisible.

Cf. A118914, A124010, A285572, A285573, A303362, A304713, A316475, A317102, A317616.

Select[Range[100],Select[Tuples[Last/@FactorInteger[#],2],And[UnsameQ@@#,Divisible@@#]&]=={}&]

A326082 Number of maximal sets of pairwise indivisible divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 5, 2, 3, 3, 5, 2, 5, 2, 5, 3, 3, 2, 8, 3, 3, 4, 5, 2, 7, 2, 6, 3, 3, 3, 9, 2, 3, 3, 8, 2, 7, 2, 5, 5, 3, 2, 12, 3, 5, 3, 5, 2, 8, 3, 8, 3, 3, 2, 15, 2, 3, 5, 7, 3, 7, 2, 5, 3, 7, 2, 15, 2, 3, 5, 5, 3, 7, 2, 12, 5, 3, 2, 15, 3

Offset: 1

Gus Wiseman, Jun 05 2019

nonn

Depends only on prime signature.

The non-maximal case is A096827.

			The maximal sets of pairwise indivisible divisors of n = 1, 2, 4, 8, 12, 24, 30, 32, 36, 48, 60 are:
   1   1   1   1   1     1      1         1    1       1       1
       2   2   2   12    24     30        2    36      48      60
           4   4   2,3   2,3    5,6       4    2,3     2,3     2,15
               8   3,4   3,4    2,15      8    2,9     3,4     3,20
                   4,6   3,8    3,10      16   3,4     3,8     4,30
                         4,6    2,3,5     32   4,18    4,6     5,12
                         6,8    6,10,15        9,12    6,8     2,3,5
                         8,12                  12,18   3,16    3,4,5
                                               4,6,9   6,16    4,5,6
                                                       8,12    3,4,10
                                                       12,16   6,15,20
                                                       16,24   10,12,15
                                                               12,15,20
                                                               12,20,30
                                                               4,6,10,15

Cf. A001055, A051026, A067992, A096827, A143824, A285572, A285573, A303362, A305148, A305149, A316476, A325861, A326023, A326077.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Rest[Subsets[Divisors[n]]],stableQ[#,Divisible]&]]],{n,100}]

A328678 Number of strict, pairwise indivisible, relatively prime integer partitions of n.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 2, 1, 2, 2, 4, 3, 5, 4, 5, 7, 10, 9, 12, 11, 14, 15, 22, 20, 25, 26, 32, 33, 44, 41, 54, 49, 62, 67, 80, 80, 100, 100, 118, 121, 152, 148, 179, 178, 210, 219, 267, 259, 316, 313, 363, 380, 449, 448, 529, 532, 619, 640, 745, 749, 867, 889

Offset: 1

Gus Wiseman, Oct 30 2019

nonn

Note that pairwise indivisibility implies strictness, but we include "strict" in the name in order to more clearly distinguish it from A328676 = "Number of relatively prime integer partitions of n whose distinct parts are pairwise indivisible".

			The a(1) = 1 through a(20) = 11 partitions (A..H = 10..20) (empty columns not shown):
  1  32  43  53  54  73   65  75   76  95   87   97   98    B7   A9    B9
         52      72  532  74  543  85  B3   B4   B5   A7    D5   B8    D7
                          83  732  94  743  D2   D3   B6    765  C7    H3
                          92       A3  752  654  754  C5    873  D6    875
                                   B2       753  853  D4    954  E5    965
                                                 952  E3    972  F4    974
                                                 B32  F2    B43  G3    A73
                                                      764   B52  H2    B54
                                                      A43   D32  865   B72
                                                      7532       964   D43
                                                                 B53   D52
                                                                 7543

The Heinz numbers of these partitions are the squarefree terms of A328677.
The non-strict case is A328676.
Pairwise indivisible partitions are A303362.
Strict, relatively prime partitions are A078374.
A ranking function using binary indices is A328671.
Cf. A000837, A285572, A285573, A304713, A305148, A316476, A328171.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&GCD@@#==1&&stableQ[#,Divisible]&]],{n,30}]

Moebius transform of A303362.

A329366 Numbers whose distinct prime indices are pairwise indivisible (stable) and pairwise non-relatively prime (intersecting).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 91, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197

Offset: 1

Gus Wiseman, Nov 12 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). A partition with no two distinct parts divisible is said to be stable, and a partition with no two distinct parts relatively prime is said to be intersecting, so these are Heinz numbers of stable intersecting partitions.

			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}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}
   37: {12}

Intersection of A316476 and A328867.
Heinz numbers of the partitions counted by A328871.
Replacing "intersecting" with "relatively prime" gives A328677.
Cf. A056239, A112798, A285573, A289509, A303362, A304713, A327393, A328671.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[100],stableQ[Union[primeMS[#]],GCD[#1,#2]==1&]&&stableQ[Union[primeMS[#]],Divisible]&]

A328871 Number of integer partitions of n whose distinct parts are pairwise indivisible (stable) and pairwise non-relatively prime (intersecting).

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 2, 4, 3, 5, 2, 6, 2, 7, 5, 7, 2, 10, 2, 11, 7, 14, 2, 16, 4, 19, 8, 22, 2, 30, 3, 29, 14, 37, 8, 48, 4, 50, 19, 59, 5, 82, 4, 81, 28, 93, 8, 128, 9, 128, 38, 147, 8, 199, 19, 196, 52, 223, 12, 308

Offset: 0

Gus Wiseman, Nov 12 2019

nonn

A partition with no two distinct parts divisible is said to be stable, and a partition with no two distinct parts relatively prime is said to be intersecting, so these are just stable intersecting partitions.

			The a(1) = 1 through a(10) = 5 partitions (A = 10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    11111  33      1111111  44        333        55
              1111         222              2222      111111111  64
                           111111           11111111             22222
                                                                 1111111111

The Heinz numbers of these partitions are A329366.
Replacing "intersecting" with "relatively prime" gives A328676.
Stable partitions are A305148.
Intersecting partitions are A328673.
Cf. A000837, A285573, A303362, A305148, A316476, A328671, A328677.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],stableQ[Union[#],Divisible]&&stableQ[Union[#],GCD[#1,#2]==1&]&]],{n,0,30}]

cp's OEIS Frontend

A318746 Number of Lyndon compositions (aperiodic necklaces of positive integers) with sum n and successive parts (including the last with the first part) being indivisible.

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A318747 Number of Lyndon compositions (aperiodic necklaces of positive integers) with sum n and adjacent parts (including the last with the first part) being indivisible (either way).

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A319496 Numbers whose prime indices are distinct and pairwise indivisible and whose own prime indices are connected and span an initial interval of positive integers.

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A371179 Positive integers with fewer distinct prime factors (A001221) than distinct divisors of prime indices (A370820).

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Formula

A317101 Numbers whose prime multiplicities are pairwise indivisible.

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A326082 Number of maximal sets of pairwise indivisible divisors of n.

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A328678 Number of strict, pairwise indivisible, relatively prime integer partitions of n.

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A329366 Numbers whose distinct prime indices are pairwise indivisible (stable) and pairwise non-relatively prime (intersecting).

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Mathematica

A328871 Number of integer partitions of n whose distinct parts are pairwise indivisible (stable) and pairwise non-relatively prime (intersecting).

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