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A318149
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e-numbers of free pure symmetric multifunctions with one atom.
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5
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1, 4, 16, 36, 128, 256, 441, 1296, 2025, 16384, 21025, 65536, 77841, 194481, 220900, 279936, 1679616, 1803649, 4100625, 4338889, 268435456, 273571600, 442050625, 449482401, 1801088541, 4294967296, 4334247225, 6059221281
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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 orderless expression e(n) (as can be represented in functional programming languages such as Mathematica) 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). The sequence consists of all numbers n such that e(n) contains no empty subexpressions f[].
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LINKS
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EXAMPLE
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The sequence of free pure symmetric multifunctions with one atom "o", together with their e-numbers begins:
1: o
4: o[o]
16: o[o,o]
36: o[o][o]
128: o[o[o]]
256: o[o,o,o]
441: o[o,o][o]
1296: o[o][o,o]
2025: o[o][o][o]
16384: o[o,o[o]]
21025: o[o[o]][o]
65536: o[o,o,o,o]
77841: o[o,o,o][o]
194481: o[o,o][o,o]
220900: o[o,o][o][o]
279936: 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]]]]]];
Select[Range[nn], FreeQ[exp[#], _[]]&]
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PROG
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(Python) See Neder link.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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