OFFSET
1,2
COMMENTS
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).
EXAMPLE
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].
MATHEMATICA
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 24 2016
STATUS
approved