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A279944
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Number of positions in the free pure symmetric multifunction in one symbol with j-number n.
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15
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1, 3, 5, 5, 7, 7, 9, 4, 7, 9, 11, 6, 9, 11, 13, 7, 8, 11, 13, 15, 9, 10, 13, 15, 9, 17, 6, 11, 12, 15, 17, 6, 11, 19, 8, 9, 13, 14, 17, 19, 8, 13, 21, 10, 11, 15, 16, 19, 11, 21, 10, 15, 23, 12, 13, 17, 18, 21, 13, 23, 12, 17, 25, 7, 14, 15, 19, 20, 23, 15, 25, 14, 19, 27, 9, 16, 17, 21, 22, 25, 9, 17, 27, 16, 21, 29, 11, 18, 19, 23, 24, 27, 11, 19, 29, 18, 23, 31, 13, 11
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OFFSET
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1,2
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COMMENTS
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A free pure symmetric multifunction in one symbol f in PSM(x) is either (case 1) f = the symbol x, or (case 2) f = an expression of the form h[g_1,...,g_k] where h is in PSM(x), each of the g_i for i=1..(k>0) is in PSM(x), and for i < j we have g_i <= g_j under a canonical total ordering of PSM(x), such as the Mathematica ordering of expressions. For a positive integer n we define a free pure symmetric multifunction j(n) by: j(1)=x; j(n>1) = j(h)[j(g_1),...,j(g_k)] where n = r(h)^(p(g_1)*...*p(g_k)-1). Here r(n) is the n-th number that is not a perfect power (A007916) and p(n) is the n-th prime number (A000040). See example. Then a(n) is the number of brackets [...] plus the number of x's in j(n).
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LINKS
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FORMULA
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EXAMPLE
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The first 20 free pure symmetric multifunctions in x are:
j(1) = j(1) = x
j(2) = j(1)[j(1)] = x[x]
j(3) = j(2)[j(1)] = x[x][x]
j(4) = j(1)[j(2)] = x[x[x]]
j(5) = j(3)[j(1)] = x[x][x][x]
j(6) = j(4)[j(1)] = x[x[x]][x]
j(7) = j(5)[j(1)] = x[x][x][x][x]
j(8) = j(1)[j(1),j(1)] = x[x,x]
j(9) = j(2)[j(2)] = x[x][x[x]]
j(10) = j(6)[j(1)] = x[x[x]][x][x]
j(11) = j(7)[j(1)] = x[x][x][x][x][x]
j(12) = j(8)[j(1)] = x[x,x][x]
j(13) = j(9)[j(1)] = x[x][x[x]][x]
j(14) = j(10)[j(1)] = x[x[x]][x][x][x]
j(15) = j(11)[j(1)] = x[x][x][x][x][x][x]
j(16) = j(1)[j(3)] = x[x[x][x]]
j(17) = j(12)[j(1)] = x[x,x][x][x]
j(18) = j(13)[j(1)] = x[x][x[x]][x][x]
j(19) = j(14)[j(1)] = x[x[x]][x][x][x][x]
j(20) = j(15)[j(1)] = x[x][x][x][x][x][x][x].
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MATHEMATICA
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nn=100;
radQ[n_]:=If[n===1, False, SameQ[GCD@@FactorInteger[n][[All, 2]], 1]];
rad[n_]:=rad[n]=If[n===0, 1, NestWhile[#+1&, rad[n-1]+1, Not[radQ[#]]&]];
Set@@@Array[radPi[rad[#]]==#&, nn];
jfac[n_]:=With[{g=GCD@@FactorInteger[n+1][[All, 2]]}, JIX[radPi[Power[n+1, 1/g]], Flatten[Cases[FactorInteger[g+1], {p_, k_}:>ConstantArray[PrimePi[p], k]]]]];
diwt[n_]:=If[n===1, 1, Apply[1+diwt[#1]+Total[diwt/@#2]&, jfac[n-1]]];
Array[diwt, nn]
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CROSSREFS
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Cf. A279984 (numbers j(n)[x]=j(prime(n))), A277576 (numbers j(n)=x[x][x][x]...), A058891 (numbers j(n)=x[x,...,x]), A279969 (numbers j(n)=x[x[...[x]]]).
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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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