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A371286
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Products of elements of A275700 (Heinz numbers of divisor sets). Numbers with a (necessarily unique) factorization into elements of A275700.
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6
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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)
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MATHEMATICA
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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]
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CROSSREFS
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A001221 counts distinct prime factors.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741 counts choices of a prime factor of each prime index.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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