A304713 -id:A304713 - cp's OEIS frontend

A318401 Numbers whose prime indices are distinct and pairwise indivisible and whose own prime indices span an initial interval of positive integers.

1, 2, 3, 7, 13, 15, 19, 35, 37, 53, 61, 69, 89, 91, 95, 113, 131, 141, 143, 145, 151, 161, 165, 223, 247, 251, 265, 281, 299, 309, 311, 329, 355, 359, 377, 385, 407, 427, 437, 463, 503, 591, 593, 611, 655, 659, 667, 671, 689, 703, 719, 721, 759, 791, 827, 851

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 strict antichains of multisets spanning an initial interval of positive integers.

			The sequence of multisystems whose MM-numbers belong to the sequence begins:
   1: {}
   2: {{}}
   3: {{1}}
   7: {{1,1}}
  13: {{1,2}}
  15: {{1},{2}}
  19: {{1,1,1}}
  35: {{2},{1,1}}
  37: {{1,1,2}}
  53: {{1,1,1,1}}
  61: {{1,2,2}}
  69: {{1},{2,2}}
  89: {{1,1,1,2}}
  91: {{1,1},{1,2}}
  95: {{2},{1,1,1}}

Cf. A003963, A006126, A055932, A056239, A112798, A285572, A290103, A293993, A302242, A304713, A316476, A319496, A319721, A319837, A320275, A320456.

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]];
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]]&]

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]&]

cp's OEIS Frontend

A318401 Numbers whose prime indices are distinct and pairwise indivisible and whose own prime indices span an initial interval of positive integers.

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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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