A303362 -id:A303362 - cp's OEIS frontend

A328671 Numbers whose binary indices are relatively prime and pairwise indivisible.

1, 6, 12, 18, 20, 22, 24, 28, 48, 56, 66, 68, 70, 72, 76, 80, 82, 84, 86, 88, 92, 96, 104, 112, 120, 132, 144, 148, 176, 192, 196, 208, 212, 224, 240, 258, 264, 272, 274, 280, 296, 304, 312, 320, 322, 328, 336, 338, 344, 352, 360, 368, 376, 384, 400, 416, 432

Offset: 1

Gus Wiseman, Oct 29 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The sequence of terms together with their binary expansions and binary indices begins:
    1:         1 ~ {1}
    6:       110 ~ {2,3}
   12:      1100 ~ {3,4}
   18:     10010 ~ {2,5}
   20:     10100 ~ {3,5}
   22:     10110 ~ {2,3,5}
   24:     11000 ~ {4,5}
   28:     11100 ~ {3,4,5}
   48:    110000 ~ {5,6}
   56:    111000 ~ {4,5,6}
   66:   1000010 ~ {2,7}
   68:   1000100 ~ {3,7}
   70:   1000110 ~ {2,3,7}
   72:   1001000 ~ {4,7}
   76:   1001100 ~ {3,4,7}
   80:   1010000 ~ {5,7}
   82:   1010010 ~ {2,5,7}
   84:   1010100 ~ {3,5,7}
   86:   1010110 ~ {2,3,5,7}
   88:   1011000 ~ {4,5,7}

The version for prime indices (instead of binary indices) is A328677.
Numbers whose binary indices are relatively prime are A291166.
Numbers whose distinct prime indices are pairwise indivisible are A316476.
BII-numbers of antichains are A326704.
Relatively prime partitions whose distinct parts are pairwise indivisible are A328676, with strict case A328678.
Cf. A000120, A121016, A285573, A303362, A304713, A305148, A316468, A316475.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[100],GCD@@bpe[#]==1&&stableQ[bpe[#],Divisible]&]

Intersection of A291166 with A326704.

A371294 Numbers whose binary indices are connected and pairwise indivisible, where two numbers are connected iff they have a common factor. A hybrid ranking sequence for connected antichains of multisets.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 40, 64, 128, 160, 256, 288, 296, 416, 512, 520, 544, 552, 640, 672, 800, 808, 928, 1024, 2048, 2176, 2304, 2432, 2560, 2688, 2816, 2944, 4096, 8192, 8200, 8224, 8232, 8320, 8352, 8480, 8488, 8608, 8704, 8712, 8736, 8744, 8832, 8864, 8992

Offset: 1

Gus Wiseman, Mar 28 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.

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The terms together with their prime indices of binary indices begin:
    1: {{}}
    2: {{1}}
    4: {{2}}
    8: {{1,1}}
   16: {{3}}
   32: {{1,2}}
   40: {{1,1},{1,2}}
   64: {{4}}
  128: {{1,1,1}}
  160: {{1,2},{1,1,1}}
  256: {{2,2}}
  288: {{1,2},{2,2}}
  296: {{1,1},{1,2},{2,2}}
  416: {{1,2},{1,1,1},{2,2}}
  512: {{1,3}}
  520: {{1,1},{1,3}}
  544: {{1,2},{1,3}}
  552: {{1,1},{1,2},{1,3}}
  640: {{1,1,1},{1,3}}
  672: {{1,2},{1,1,1},{1,3}}
  800: {{1,2},{2,2},{1,3}}
  808: {{1,1},{1,2},{2,2},{1,3}}
  928: {{1,2},{1,1,1},{2,2},{1,3}}

Connected case of A087086, relatively prime A328671.
For binary indices of binary indices we have A326750, non-primitive A326749.
For prime indices of prime indices we have A329559, non-primitive A305078.
Primitive case of A371291 = positions of ones in A371452.
For binary indices of prime indices we have A371445, non-primitive A325118.
A001187 counts connected graphs.
A007718 counts non-isomorphic connected multiset partitions.
A048143 counts connected antichains of sets.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326964 counts connected set-systems, covering A323818.
Cf. A001222, A051026, A285572, A303362, A304713, A305079, A316476, A319496, A319719, A326704, A371446.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],stableQ[bpe[#],Divisible]&&connectedQ[prix/@bpe[#]]&]

Intersection of A087086 and A371291.

A327393 Maximum stable divisor of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 4, 13, 7, 15, 16, 17, 9, 19, 5, 7, 11, 23, 8, 25, 13, 27, 7, 29, 15, 31, 32, 33, 17, 35, 9, 37, 19, 13, 8, 41, 7, 43, 11, 45, 23, 47, 16, 49, 25, 51, 13, 53, 27, 55, 8, 19, 29, 59, 15, 61, 31, 9, 64, 13, 33, 67, 17, 69, 35, 71

Offset: 1

Gus Wiseman, Sep 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. A number is stable if its distinct prime indices are pairwise indivisible. Stable numbers are listed in A316476, which is the union of this sequence without 1.

			The stable divisors of 60 are {1, 2, 3, 4, 5, 15}, so a(60) = 15.

Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional cross-references.

Cf. A000005, A006530, A302242, A303362.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Max[Select[Divisors[n],stableQ[PrimePi/@First/@FactorInteger[#],Divisible]&]],{n,100}]

A322438 Number of unordered pairs of factorizations of n into factors > 1 where no factor of one properly divides any factor of the other.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 4

Offset: 1

Gus Wiseman, Dec 08 2018

nonn

First differs from A322437 at a(144) = 4, A322437(144) = 3.

First differs from A379958 at a(120) = 2, A379958(120) = 1.

			The a(240) = 5 pairs of factorizations::
  (4*4*15)|(4*6*10)
    (6*40)|(15*16)
    (8*30)|(12*20)
   (10*24)|(15*16)
   (12*20)|(15*16)

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n.

Cf. A001055, A285572, A285573, A303362, A303386, A304713, A305149, A305150, A305193, A316476, A317144, A322435, A322436, A322437, A322442, A379958.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
divpropQ[x_,y_]:=And[x!=y,Divisible[x,y]];
Table[Length[Select[Subsets[facs[n],{2}],And[!Or@@divpropQ@@@Tuples[#],!Or@@divpropQ@@@Reverse/@Tuples[#]]&]],{n,100}]

factorizations(n, m=n, f=List([]), z=List([])) = if(1==n, listput(z,Vec(f)); z, my(newf); fordiv(n, d, if((d>1)&&(d<=m), newf = List(f); listput(newf,d); z = factorizations(n/d, d, newf, z))); (z));
is_proper_ndf_pair(fac1,fac2) = { for(i=1,#fac1,for(j=1,#fac2,if((fac1[i]!=fac2[j]) && (!(fac1[i]%fac2[j]) || !(fac2[j]%fac1[i])),return(0)))); (1); };
number_of_proper_ndfpairs(z) = sum(i=1,#z,sum(j=i+1,#z,is_proper_ndf_pair(z[i],z[j])));
A322438(n) = number_of_proper_ndfpairs(Vec(factorizations(n))); \\ Antti Karttunen, Jan 24 2025

Data section extended up to a(144) by Antti Karttunen, Jan 24 2025

A317616 Numbers whose prime multiplicities are not pairwise indivisible.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192

Offset: 1

Gus Wiseman, Aug 01 2018

nonn

The numbers of terms that do not exceed 10^k, for k = 2, 3, ..., are 26, 344, 3762, 38711, 390527, 3915874, 39192197, 392025578, 3920580540, ... . Apparently, the asymptotic density of this sequence exists and equals 0.392... . - Amiram Eldar, Sep 25 2024

			72 = 2^3 * 3^2 is not in the sequence because 3 and 2 are pairwise indivisible.

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

Cf. A118914, A124010, A285572, A285573, A303362, A304713, A316475, A317101, A317102.

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

is(k) = if(k == 1, 0, my(e = Set(factor(k)[,2])); if(vecmax(e) == 1, 0, for(i = 1, #e, for(j = 1, i-1, if(!(e[i] % e[j]), return(1)))); 0)); \\ Amiram Eldar, Sep 25 2024

A318730 Number of cyclic compositions (necklaces of positive integers) summing to n with adjacent parts (including the last and first part) being indivisible (either way).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 3, 6, 5, 8, 7, 14, 15, 21, 31, 39, 51, 69, 98, 133, 177, 254, 329, 471, 632, 902, 1230, 1710, 2370, 3270, 4591, 6384, 8898, 12429, 17252, 24230, 33783, 47405, 66254, 92860, 130142, 182469, 256262, 359676, 505231, 710059, 997953, 1404215

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

			The a(14) = 14 cyclic 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) (2,5,2,5) (2,5,4,3) (3,4,3,4)

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

Cf. A000740, A008965, A059966, A167606, A285573, A303362, A304713, A316476, A318726, A318727, A318728, A318729, A328601.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[Length[#]==1,And[neckQ[#],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, eulerphi(d)*v[n/d])/n)} \\ Andrew Howroyd, Oct 27 2019

a(n) = A328601(n) + 1. - Andrew Howroyd, Oct 27 2019

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

A303364 Number of strict integer partitions of n with pairwise indivisible and squarefree parts.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 2, 1, 1, 3, 2, 2, 4, 3, 3, 4, 6, 5, 5, 6, 7, 8, 9, 10, 10, 11, 11, 14, 14, 17, 16, 18, 19, 23, 24, 27, 29, 30, 33, 36, 41, 41, 42, 46, 51, 56, 60, 66, 67, 71, 81, 86, 93, 96, 101, 110, 121, 129, 135, 144, 153, 159, 173, 192, 204, 207, 224

Offset: 1

Gus Wiseman, Apr 22 2018

nonn

			The a(23) = 9 strict integer partitions are (23), (13,10), (17,6), (21,2), (10,7,6), (11,7,5), (13,7,3), (11,7,3,2), (13,5,3,2).

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..700 (terms 0..400 from Andrew Howroyd)

Cf. A000009, A000837, A003238, A005117, A006126, A051424, A073576, A285572, A285573, A293606, A293993, A303362, A303365.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@SquareFreeQ/@#&&Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]==={}&]],{n,60}]

lista(nn)={local(Cache=Map());
  my(excl=vector(nn, n, sumdiv(n, d, 2^(n-d))));
  my(c(n, m, b)=
     if(n==0, 1,
        while(m>n || bittest(b,0), m--; b>>=1);
        my(hk=[n, m, b], z);
        if(!mapisdefined(Cache, hk, &z),
          z = if(m, self()(n, m-1, b>>1) + self()(n-m, m, bitor(b, excl[m])), 0);
          mapput(Cache, hk, z)); z));
  my(a(n)=c(n, n, sum(i=1, n, if(!issquarefree(i), 2^(n-i)))));
  for(n=1, nn, print1(a(n), ", "))
} \\ Andrew Howroyd, Nov 02 2019

A305081 Heinz numbers of z-trees. Heinz numbers of connected integer partitions with pairwise indivisible parts and z-density -1.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 91, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 203, 211, 223, 227, 229

Offset: 1

Gus Wiseman, May 25 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A multiset S is said to be connected if G(S) is a connected graph.
The clutter density of a multiset S of positive integers is Sum_{s in S} (omega(s) - 1) - omega(lcm(S)) where omega = A001221.

			4331 is the Heinz number of {18,20}, which is a z-tree corresponding to the multiset multisystem {{1,2,2},{1,1,3}}.
17927 is the Heinz number of {4,6,45}, which is a z-tree corresponding to the multiset multisystem {{1,1},{1,2},{2,2,3}}.
27391 is the Heinz number of {4,4,6,14}, which is a z-tree corresponding to the multiset multisystem {{1,1},{1,1},{1,2},{1,4}}.

Cf. A001221, A030019, A056239, A112798, A286520, A302242, A303362, A303837, A304118, A304714, A304716, A305052, A305078, A305079.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
zens[n_]:=If[n==1,0,Total@Cases[FactorInteger[n],{p_,k_}:>k*(PrimeNu[PrimePi[p]]-1)]-PrimeNu[LCM@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]]]];
Select[Range[300],And[zens[#]==-1,Length[zsm[primeMS[#]]]==1,Select[Tuples[primeMS[#],2],UnsameQ@@#&&Divisible@@#&]=={}]&]

A328676 Number of relatively prime integer partitions of n whose distinct parts are pairwise indivisible.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 4, 3, 5, 5, 11, 7, 16, 14, 18, 22, 34, 30, 47, 45, 59, 66, 89, 90, 118, 125, 159, 169, 218, 225, 289, 304, 369, 400, 486, 520, 636, 680, 806, 873, 1051, 1105, 1333, 1424, 1664, 1803, 2122, 2253, 2659, 2841, 3283, 3560, 4118, 4388, 5096

Offset: 1

Gus Wiseman, Oct 29 2019

nonn

			The a(4) = 1 through a(11) = 11 partitions:
  1111  32     111111  43       53        54         73          65
        11111          52       332       72         433         74
                       322      11111111  522        532         83
                       1111111            3222       3322        92
                                          111111111  1111111111  443
                                                                 533
                                                                 722
                                                                 3332
                                                                 5222
                                                                 32222
                                                                 11111111111

The Heinz numbers of these partitions are given by A328677.
The strict case is A328678.
The binary index version is A328671.
Relatively prime partitions are A000837.
Partitions whose distinct parts are pairwise indivisible are A305148.
Cf. A285573, A289509, A303362, A316476, A326912, A327393, A328171.

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

A329626 Smallest BII-number of an antichain with n edges.

Original entry on oeis.org

0, 1, 3, 11, 139, 820, 2868, 35636, 199476, 723764

Offset: 0

Gus Wiseman, Nov 28 2019

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets of positive integers) has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

A set-system is an antichain if no edge is a proper subset of any other.

			The sequence of terms together with their corresponding set-systems begins:
       0: {}
       1: {{1}}
       3: {{1},{2}}
      11: {{1},{2},{3}}
     139: {{1},{2},{3},{4}}
     820: {{1,2},{1,3},{2,3},{1,4},{2,4}}
    2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
   35636: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{5}}
  199476: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5},{2,5}}
  723764: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5},{2,5},{3,5}}

The connected case is A329627.
The intersecting case is A329628.
BII-numbers of antichains are A326704.
Antichain covers are A006126.
Cf. A048143, A048793, A070939, A303362, A319837, A326031, A326750, A329555, A329560, A329561, A329625.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_]:=!Apply[Or,Outer[#1=!=#2&&SubsetQ[#1,#2]&,u,u,1],{0,1}];
First/@GatherBy[Select[Range[0,10000],stableQ[bpe/@bpe[#]]&],Length[bpe[#]]&]

cp's OEIS Frontend

A328671 Numbers whose binary indices are relatively prime and pairwise indivisible.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A371294 Numbers whose binary indices are connected and pairwise indivisible, where two numbers are connected iff they have a common factor. A hybrid ranking sequence for connected antichains of multisets.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A327393 Maximum stable divisor of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A322438 Number of unordered pairs of factorizations of n into factors > 1 where no factor of one properly divides any factor of the other.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A317616 Numbers whose prime multiplicities are not pairwise indivisible.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A318730 Number of cyclic compositions (necklaces of positive integers) summing to n with adjacent parts (including the last and first part) being indivisible (either way).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A303364 Number of strict integer partitions of n with pairwise indivisible and squarefree parts.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A305081 Heinz numbers of z-trees. Heinz numbers of connected integer partitions with pairwise indivisible parts and z-density -1.

Views

Author

Keywords