A259936 -id:A259936 - cp's OEIS frontend

A305106 Number of unitary factorizations of Heinz numbers of integer partitions of n. Number of multiset partitions of integer partitions of n with pairwise disjoint blocks.

1, 1, 2, 4, 7, 12, 21, 34, 55, 87, 138, 211, 324, 486, 727, 1079, 1584, 2305, 3337, 4789, 6830, 9712, 13689, 19225, 26841, 37322, 51598, 71108, 97580, 133350, 181558, 246335, 332991, 448706, 602607, 806732, 1077333, 1433885, 1903682, 2520246, 3328549, 4383929

Offset: 0

Gus Wiseman, May 25 2018

nonn

			The a(6) = 21 unitary factorizations:
(13) (21) (22) (25) (27) (28) (30) (36) (40) (48) (64)
(2*11) (2*15) (3*7) (3*10) (3*16) (4*7) (4*9) (5*6) (5*8)
(2*3*5)
The a(6) = 21 multiset partitions:
{{6}}
{{2,4}}
{{1,5}}
{{3,3}}
{{2,2,2}}
{{1,1,4}}
{{1,2,3}}
{{1,1,2,2}}
{{1,1,1,3}}
{{1,1,1,1,2}}
{{1,1,1,1,1,1}}
{{1},{5}}
{{1},{2,3}}
{{2},{4}}
{{2},{1,3}}
{{2},{1,1,1,1}}
{{1,1},{4}}
{{1,1},{2,2}}
{{3},{1,2}}
{{3},{1,1,1}}
{{1},{2},{3}}

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000110, A001055, A001221, A001970, A034444, A089233, A258466, A259936, A281116, A285572, A305078, A305079.

Row sums of A321878.

Table[Sum[BellB[Length[Union[y]]],{y,IntegerPartitions[n]}],{n,30}]
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[With[{t = n - i j}, b[t, Min[t, i - 1], k]], {j, 1, n/i}] k + b[n, i - 1, k]]];
T[n_, k_] := Sum[b[n, n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]/k!;
a[n_] := Sum[T[n, k], {k, 0, Floor[(Sqrt[1 + 8n] - 1)/2]}];
a /@ Range[0, 50] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz in A321878 *)

A327521 Number of factorizations of the n-th squarefree number A005117(n) into squarefree numbers > 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 5, 1, 2, 2, 2, 1, 2, 2, 1, 5, 1, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 5, 1, 2, 5, 1, 1, 2, 2, 5, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 1, 5, 1, 5, 2, 1, 1, 5, 2, 1, 5, 2, 2, 2, 2, 2, 1, 2, 5, 1, 2, 2, 1, 5, 1, 2

Offset: 1

Gus Wiseman, Sep 15 2019

nonn

Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional cross-references.

Cf. A000110, A005117, A007947, A034444, A050320, A072047, A259936.

nn=100;
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
y=Select[Range[nn],SquareFreeQ];
Table[Length[facsusing[Rest[y],n]],{n,y}]

a(n) = A050320(A005117(n)) = A000110(A072047(n)) = A000110(A001222(A005117(n))).

A327905 FDH numbers of pairwise coprime sets.

Original entry on oeis.org

2, 6, 8, 10, 12, 14, 18, 20, 21, 22, 24, 26, 28, 32, 33, 34, 35, 38, 40, 42, 44, 46, 48, 50, 52, 55, 56, 57, 58, 62, 63, 66, 68, 70, 74, 75, 76, 77, 80, 82, 84, 86, 88, 91, 93, 94, 95, 96, 98, 99, 100, 104, 106, 110, 112, 114, 116, 118, 122, 123, 125, 126, 132

Offset: 1

Gus Wiseman, Sep 30 2019

nonn

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. The FDH-number of a strict partition or finite set {y_1,...,y_k} is f(y_1)*...*f(y_k).

We use the Mathematica function CoprimeQ, meaning a singleton is not coprime unless it is {1}.

			The sequence of terms together with their corresponding coprime sets begins:
   2: {1}
   6: {1,2}
   8: {1,3}
  10: {1,4}
  12: {2,3}
  14: {1,5}
  18: {1,6}
  20: {3,4}
  21: {2,5}
  22: {1,7}
  24: {1,2,3}
  26: {1,8}
  28: {3,5}
  32: {1,9}
  33: {2,7}
  34: {1,10}
  35: {4,5}
  38: {1,11}
  40: {1,3,4}
  42: {1,2,5}

Wolfram Language Documentation, CoprimeQ

Heinz numbers of pairwise coprime partitions are A302696 (all), A302797 (strict), A302569 (with singletons), and A302798 (strict with singletons).
FDH numbers of relatively prime sets are A319827.
Cf. A050376, A056239, A064547, A213925, A259936, A299755, A299757, A304711, A319826, A326675.

FDfactor[n_]:=If[n==1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}->2^(m-1)]]]]];
nn=100;FDprimeList=Array[FDfactor,nn,1,Union];
FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
Select[Range[nn],CoprimeQ@@(FDfactor[#]/.FDrules)&]

cp's OEIS Frontend

A305106 Number of unitary factorizations of Heinz numbers of integer partitions of n. Number of multiset partitions of integer partitions of n with pairwise disjoint blocks.

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A327521 Number of factorizations of the n-th squarefree number A005117(n) into squarefree numbers > 1.

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A327905 FDH numbers of pairwise coprime sets.

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