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A368110
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Numbers of which it is possible to choose a different divisor of each prime index.
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41
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1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97
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OFFSET
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1,2
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COMMENTS
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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.
By Hall's marriage theorem, k is a term if and only if there is no sub-multiset S of the prime indices of k such that fewer than |S| numbers are divisors of a member of S. Equivalently, there is no divisor of k in A370348. - Robert Israel, Feb 15 2024
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LINKS
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FORMULA
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Heinz numbers of the partitions counted by A239312.
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EXAMPLE
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The terms together with their prime indices begin:
1: {}
2: {1}
3: {2}
5: {3}
6: {1,2}
7: {4}
9: {2,2}
10: {1,3}
11: {5}
13: {6}
14: {1,4}
15: {2,3}
17: {7}
19: {8}
21: {2,4}
22: {1,5}
23: {9}
25: {3,3}
26: {1,6}
29: {10}
30: {1,2,3}
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MAPLE
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filter:= proc(n) uses numtheory, GraphTheory; local B, S, F, D, E, G, t, d;
F:= ifactors(n)[2];
F:= map(t -> [pi(t[1]), t[2]], F);
D:= `union`(seq(divisors(t[1]), t = F));
F:= map(proc(t) local i; seq([t[1], i], i=1..t[2]) end proc, F);
if nops(D) < nops(F) then return false fi;
E:= {seq(seq({t, d}, d=divisors(t[1])), t = F)};
S:= map(t -> convert(t, name), [op(F), op(D)]);
E:= map(e -> map(convert, e, name), E);
G:= Graph(S, E);
B:= BipartiteMatching(G);
B[1] = nops(F);
end proc:
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MATHEMATICA
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prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Select[Tuples[Divisors/@prix[#]], UnsameQ@@#&]!={}&]
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CROSSREFS
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Positions of nonzero terms in A355739.
A003963 multiplies together the prime indices of n.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A000720, A076610, A111774, A335433, A335448, A340852, A355733, A355734, A355737, A355749, A370348.
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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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