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

%I A371286 #7 Mar 22 2024 16:28:07
%S A371286 1,2,4,6,8,10,12,16,20,22,24,32,34,36,40,42,44,48,60,62,64,68,72,80,
%T A371286 82,84,88,96,100,118,120,124,128,132,134,136,144,160,164,166,168,176,
%U A371286 192,200,204,216,218,220,230,236,240,248,252,254,256,264,268,272,288
%N A371286 Products of elements of A275700 (Heinz numbers of divisor sets). Numbers with a (necessarily unique) factorization into elements of A275700.
%C A371286 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.
%C A371286 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.
%e A371286 The terms together with their prime factorizations and unique factorizations into terms of A275700 begin:
%e A371286    1 =             = ()
%e A371286    2 = 2           = (2)
%e A371286    4 = 2*2         = (2*2)
%e A371286    6 = 2*3         = (6)
%e A371286    8 = 2*2*2       = (2*2*2)
%e A371286   10 = 2*5         = (10)
%e A371286   12 = 2*2*3       = (2*6)
%e A371286   16 = 2*2*2*2     = (2*2*2*2)
%e A371286   20 = 2*2*5       = (2*10)
%e A371286   22 = 2*11        = (22)
%e A371286   24 = 2*2*2*3     = (2*2*6)
%e A371286   32 = 2*2*2*2*2   = (2*2*2*2*2)
%e A371286   34 = 2*17        = (34)
%e A371286   36 = 2*2*3*3     = (6*6)
%e A371286   40 = 2*2*2*5     = (2*2*10)
%e A371286   42 = 2*3*7       = (42)
%e A371286   44 = 2*2*11      = (2*22)
%e A371286   48 = 2*2*2*2*3   = (2*2*2*6)
%e A371286   60 = 2*2*3*5     = (6*10)
%e A371286   62 = 2*31        = (62)
%e A371286   64 = 2*2*2*2*2*2 = (2*2*2*2*2*2)
%e A371286   68 = 2*2*17      = (2*34)
%e A371286   72 = 2*2*2*3*3   = (2*6*6)
%e A371286   80 = 2*2*2*2*5   = (2*2*2*10)
%e A371286   82 = 2*41        = (82)
%e A371286   84 = 2*2*3*7     = (2*42)
%e A371286   88 = 2*2*2*11    = (2*2*22)
%e A371286   96 = 2*2*2*2*2*3 = (2*2*2*2*6)
%t A371286 nn=100;
%t A371286 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A371286 facs[n_]:=If[n<=1, {{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]], {d,Rest[Divisors[n]]}]];
%t A371286 s=Table[Times@@Prime/@Divisors[n],{n,nn}];
%t A371286 m=Max@@Table[Select[Range[2,k],prix[#] == Divisors[Last[prix[#]]]&],{k,nn}];
%t A371286 Join@@Position[Table[Length[Select[facs[n], SubsetQ[s,Union[#]]&]],{n,m}],1]
%Y A371286 Products of elements of A275700.
%Y A371286 The squarefree case is A371283.
%Y A371286 The unsorted version is A371285.
%Y A371286 A000005 counts divisors.
%Y A371286 A001221 counts distinct prime factors.
%Y A371286 A027746 lists prime factors, A112798 indices, length A001222.
%Y A371286 A355731 counts choices of a divisor of each prime index, firsts A355732.
%Y A371286 A355741 counts choices of a prime factor of each prime index.
%Y A371286 Cf. A000720, A003963, A005179, A007416, A034729, A054973, A056239, A370820, A371284, A371288.
%K A371286 nonn
%O A371286 1,2
%A A371286 _Gus Wiseman_, Mar 22 2024