A319782 -id:A319782 - cp's OEIS frontend

A319786 Number of factorizations of n where no two factors are relatively prime.

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 4, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 7, 2, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 3, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 7, 1, 1, 2, 2, 1, 1, 1, 7, 5, 1, 1, 3, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 12, 1, 2, 2, 4, 1, 1, 1, 4, 1

Offset: 1

Gus Wiseman, Sep 27 2018

nonn

First differs from A305193 at a(36) = 4, A305193(36) = 5.

a(n) depends only on prime signature of n (cf. A025487). - Antti Karttunen, Nov 07 2018

			The a(48) = 7 factorizations are (2*2*2*6), (2*2*12), (2*4*6), (2*24), (4*12), (6*8), (48).

Antti Karttunen, Table of n, a(n) for n = 1..11520
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A001055, A007716, A045778, A049311, A162247, A281116, A283877, A305193, A305854, A306006, A316980, A318749.

Cf. A319755, A319759, A319760, A319765, A319779, A319787, A319782, A319789.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],!Or@@CoprimeQ@@@Subsets[#,{2}]&]],{n,100}]

A319786(n, m=n, facs=List([])) = if(1==n, (1!=gcd(Vec(facs))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A319786(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Nov 07 2018

More terms from Antti Karttunen, Nov 07 2018

A319787 Number of intersecting multiset partitions of normal multisets of size n.

Original entry on oeis.org

1, 1, 3, 8, 27, 95, 373, 1532, 6724

Offset: 0

Gus Wiseman, Sep 27 2018

A multiset is normal if it spans an initial interval of positive integers.

A multiset partition is intersecting iff no two parts are disjoint.

			The a(1) = 1 through a(3) = 8 multiset partitions:
1: {{1}}
2: {{1,1}}
   {{1,2}}
   {{1},{1}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1,1,2}}
   {{1,2,3}}
   {{1},{1,1}}
   {{2},{1,2}}
   {{1},{1,2}}
   {{1},{1},{1}}

Cf. A007716, A049311, A283877, A305854, A306006, A316980, A317752.

Cf. A319755, A319759, A319760, A319765, A319779, A319787, A319782, A319784, A319789.

A319789 Number of intersecting multiset partitions of strongly normal multisets of size n.

Original entry on oeis.org

1, 1, 3, 6, 17, 40, 122, 330, 1032

Offset: 0

Gus Wiseman, Sep 27 2018

A multiset is normal if it spans an initial interval of positive integers, and strongly normal if in addition its multiplicities are weakly decreasing. A multiset partition is intersecting iff no two parts are disjoint.

			The a(1) = 1 through a(3) = 6 multiset partitions:
1: {{1}}
2: {{1,1}}
   {{1,2}}
   {{1},{1}}
3: {{1,1,1}}
   {{1,1,2}}
   {{1,2,3}}
   {{1},{1,1}}
   {{1},{1,2}}
   {{1},{1},{1}}

Cf. A007716, A283877, A305854, A306006, A316980, A317755.

Cf. A319755, A319759, A319760, A319765, A319779, A319787, A319782, A319784, A319787.

A319784 Number of non-isomorphic intersecting T_0 set systems of weight n.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 3, 5, 7, 14, 25

Offset: 0

Gus Wiseman, Sep 27 2018

A multiset partition is intersecting if no two parts are disjoint. The weight of a multiset partition is the sum of sizes of its parts. The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. The T_0 condition means the dual is strict.

			Non-isomorphic representatives of the a(1) = 1 through a(8) = 7 multiset partitions:
1: {{1}}
3: {{2},{1,2}}
4: {{1,3},{2,3}}
5: {{3},{1,3},{2,3}}
6: {{3},{2,3},{1,2,3}}
   {{1,2},{1,3},{2,3}}
   {{1,4},{2,4},{3,4}}
7: {{4},{1,3,4},{2,3,4}}
   {{1,3},{1,4},{2,3,4}}
   {{1,3},{2,3},{1,2,3}}
   {{1,4},{3,4},{2,3,4}}
   {{4},{1,4},{2,4},{3,4}}
8: {{1,5},{2,4,5},{3,4,5}}
   {{2,4},{3,4},{1,2,3,4}}
   {{2,4},{1,2,5},{3,4,5}}
   {{2,4},{1,3,4},{2,3,4}}
   {{3},{1,3},{2,3},{1,2,3}}
   {{4},{1,4},{3,4},{2,3,4}}
   {{1,5},{2,5},{3,5},{4,5}}

Cf. A007716, A049311, A283877, A305843, A305854, A306006, A316980, A317752.

Cf. A319755, A319759, A319760, A319765, A319779, A319787, A319782, A319789.

cp's OEIS Frontend

A319786 Number of factorizations of n where no two factors are relatively prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A319787 Number of intersecting multiset partitions of normal multisets of size n.

Views

Author

Keywords

Comments

Examples

Crossrefs

A319789 Number of intersecting multiset partitions of strongly normal multisets of size n.

Views

Author

Keywords

Comments

Examples

Crossrefs

A319784 Number of non-isomorphic intersecting T_0 set systems of weight n.

Views

Author

Keywords

Comments

Examples

Crossrefs