A371286 Products of elements of A275700 (Heinz numbers of divisor sets). Numbers with a (necessarily unique) factorization into elements of A275700.

1, 2, 4, 6, 8, 10, 12, 16, 20, 22, 24, 32, 34, 36, 40, 42, 44, 48, 60, 62, 64, 68, 72, 80, 82, 84, 88, 96, 100, 118, 120, 124, 128, 132, 134, 136, 144, 160, 164, 166, 168, 176, 192, 200, 204, 216, 218, 220, 230, 236, 240, 248, 252, 254, 256, 264, 268, 272, 288

Offset: 1

Author

Gus Wiseman, Mar 22 2024

Keywords

• nonn

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

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.

Examples

			The terms together with their prime factorizations and unique factorizations into terms of A275700 begin:
   1 =             = ()
   2 = 2           = (2)
   4 = 2*2         = (2*2)
   6 = 2*3         = (6)
   8 = 2*2*2       = (2*2*2)
  10 = 2*5         = (10)
  12 = 2*2*3       = (2*6)
  16 = 2*2*2*2     = (2*2*2*2)
  20 = 2*2*5       = (2*10)
  22 = 2*11        = (22)
  24 = 2*2*2*3     = (2*2*6)
  32 = 2*2*2*2*2   = (2*2*2*2*2)
  34 = 2*17        = (34)
  36 = 2*2*3*3     = (6*6)
  40 = 2*2*2*5     = (2*2*10)
  42 = 2*3*7       = (42)
  44 = 2*2*11      = (2*22)
  48 = 2*2*2*2*3   = (2*2*2*6)
  60 = 2*2*3*5     = (6*10)
  62 = 2*31        = (62)
  64 = 2*2*2*2*2*2 = (2*2*2*2*2*2)
  68 = 2*2*17      = (2*34)
  72 = 2*2*2*3*3   = (2*6*6)
  80 = 2*2*2*2*5   = (2*2*2*10)
  82 = 2*41        = (82)
  84 = 2*2*3*7     = (2*42)
  88 = 2*2*2*11    = (2*2*22)
  96 = 2*2*2*2*2*3 = (2*2*2*2*6)
		

Crossrefs

Products of elements of A275700.

The squarefree case is A371283.

The unsorted version is A371285.

A000005 counts divisors.

A001221 counts distinct prime factors.

A027746 lists prime factors, A112798 indices, length A001222.

A355731 counts choices of a divisor of each prime index, firsts A355732.

A355741 counts choices of a prime factor of each prime index.

Cf. A000720, A003963, A005179, A007416, A034729, A054973, A056239, A370820, A371284, A371288.

Programs

Mathematica

nn=100;
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1, {{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]], {d,Rest[Divisors[n]]}]];
s=Table[Times@@Prime/@Divisors[n],{n,nn}];
m=Max@@Table[Select[Range[2,k],prix[#] == Divisors[Last[prix[#]]]&],{k,nn}];
Join@@Position[Table[Length[Select[facs[n], SubsetQ[s,Union[#]]&]],{n,m}],1]