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A317765
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Number of distinct subexpressions of the free pure symmetric multifunction (with empty expressions allowed) with e-number n.
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8
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1, 2, 3, 2, 4, 3, 5, 3, 3, 4, 6, 4, 4, 5, 7, 2, 5, 5, 6, 8, 3, 6, 6, 7, 4, 9, 3, 4, 7, 7, 8, 4, 5, 10, 4, 3, 5, 8, 8, 9, 5, 6, 11, 5, 4, 6, 9, 9, 5, 10, 6, 7, 12, 6, 5, 7, 10, 10, 6, 11, 7, 8, 13, 3, 7, 6, 8, 11, 11, 7, 12, 8, 9, 14, 4, 8, 7, 9, 12, 12, 3, 8
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OFFSET
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1,2
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COMMENTS
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If n = 1 let e(n) be the leaf symbol "o". Given a positive integer n > 1 we construct a unique free pure symmetric multifunction (with empty expressions allowed) e(n) with one atom by expressing n as a power of a number that is not a perfect power to a product of prime numbers: n = rad(x)^(prime(y_1) * ... * prime(y_k)) where rad = A007916. Then e(n) = e(x)[e(y_1), ..., e(y_k)]. For example, e(21025) = o[o[o]][o] because 21025 = rad(rad(1)^prime(rad(1)^prime(1)))^prime(1).
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LINKS
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EXAMPLE
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The a(12) = 4 subexpressions of o[o[]][] are {o, o[], o[o[]], o[o[]][]}.
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MATHEMATICA
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nn=1000;
radQ[n_]:=If[n===1, False, GCD@@FactorInteger[n][[All, 2]]===1];
rad[n_]:=rad[n]=If[n===0, 1, NestWhile[#+1&, rad[n-1]+1, Not[radQ[#]]&]];
Clear[radPi]; Set@@@Array[radPi[rad[#]]==#&, nn];
exp[n_]:=If[n===1, "o", With[{g=GCD@@FactorInteger[n][[All, 2]]}, Apply[exp[radPi[Power[n, 1/g]]], exp/@Flatten[Cases[FactorInteger[g], {p_?PrimeQ, k_}:>ConstantArray[PrimePi[p], k]]]]]];
Table[Length[Union[Cases[exp[n], _, {0, Infinity}, Heads->True]]], {n, 100}]
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CROSSREFS
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Cf. A007916, A052409, A052410, A277576, A277996, A300626, A316112, A317056, A317658, A317713, A317994.
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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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