A304711 -id:A304711 - cp's OEIS frontend

A327525 Number of factorizations of A302569(n), the n-th number that is 1, prime, or whose prime indices are pairwise coprime.

1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 4, 1, 2, 2, 5, 1, 1, 4, 2, 1, 7, 2, 4, 1, 5, 1, 7, 2, 2, 2, 1, 2, 7, 1, 1, 4, 2, 1, 12, 2, 4, 1, 2, 7, 2, 1, 11, 1, 2, 11, 5, 1, 4, 2, 5, 1, 1, 2, 4, 2, 1, 12, 2, 1, 2, 2, 7, 1, 4, 2, 2, 2, 19, 1, 1, 5, 1, 7, 2, 1, 1, 5, 12, 1, 4

Offset: 1

Gus Wiseman, Sep 20 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 a(47) = 11 factorizations of 60 together with the corresponding multiset partitions of {1,1,2,3}:
  (2*2*3*5)  {{1},{1},{2},{3}}
  (2*2*15)   {{1},{1},{2,3}}
  (2*3*10)   {{1},{2},{1,3}}
  (2*5*6)    {{1},{3},{1,2}}
  (2*30)     {{1},{1,2,3}}
  (3*4*5)    {{2},{1,1},{3}}
  (3*20)     {{2},{1,1,3}}
  (4*15)     {{1,1},{2,3}}
  (5*12)     {{3},{1,1,2}}
  (6*10)     {{1,2},{1,3}}
  (60)       {{1,1,2,3}}

Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional cross-references.

Cf. A056239, A112798, A281116, A318721, A302569, A304711, A305079.

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,#]&]}]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
y=Select[Range[nn],PrimeQ[#]||CoprimeQ@@primeMS[#]&];
Table[Length[facsusing[Rest[y],n]],{n,y}]

a(n) = A001055(A302569(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

A327525 Number of factorizations of A302569(n), the n-th number that is 1, prime, or whose prime indices are pairwise coprime.

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