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%I #15 May 04 2018 22:42:22
%S 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,
%T 12,15,17,6,11,19,8,9,13,14,17,19,8,13,21,10,11,15,16,19,11,21,10,15,
%U 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
%N Number of positions in the free pure symmetric multifunction in one symbol with j-number n.
%C 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).
%F a(A007916(h)^(A000040(g_1)*...*A000040(g_k)-1)) = 1 + a(h) + a(g_1) + ... + a(g_k).
%e The first 20 free pure symmetric multifunctions in x are:
%e j(1) = j(1) = x
%e j(2) = j(1)[j(1)] = x[x]
%e j(3) = j(2)[j(1)] = x[x][x]
%e j(4) = j(1)[j(2)] = x[x[x]]
%e j(5) = j(3)[j(1)] = x[x][x][x]
%e j(6) = j(4)[j(1)] = x[x[x]][x]
%e j(7) = j(5)[j(1)] = x[x][x][x][x]
%e j(8) = j(1)[j(1),j(1)] = x[x,x]
%e j(9) = j(2)[j(2)] = x[x][x[x]]
%e j(10) = j(6)[j(1)] = x[x[x]][x][x]
%e j(11) = j(7)[j(1)] = x[x][x][x][x][x]
%e j(12) = j(8)[j(1)] = x[x,x][x]
%e j(13) = j(9)[j(1)] = x[x][x[x]][x]
%e j(14) = j(10)[j(1)] = x[x[x]][x][x][x]
%e j(15) = j(11)[j(1)] = x[x][x][x][x][x][x]
%e j(16) = j(1)[j(3)] = x[x[x][x]]
%e j(17) = j(12)[j(1)] = x[x,x][x][x]
%e j(18) = j(13)[j(1)] = x[x][x[x]][x][x]
%e j(19) = j(14)[j(1)] = x[x[x]][x][x][x][x]
%e j(20) = j(15)[j(1)] = x[x][x][x][x][x][x][x].
%t nn=100;
%t radQ[n_]:=If[n===1,False,SameQ[GCD@@FactorInteger[n][[All,2]],1]];
%t rad[n_]:=rad[n]=If[n===0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];
%t Set@@@Array[radPi[rad[#]]==#&,nn];
%t 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]]]]];
%t diwt[n_]:=If[n===1,1,Apply[1+diwt[#1]+Total[diwt/@#2]&,jfac[n-1]]];
%t Array[diwt,nn]
%Y Cf. A005043, A007916, A106490, A277564, A277615, A277996, A278028, A280000.
%Y 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]]]).
%K nonn
%O 1,2
%A _Gus Wiseman_, Dec 24 2016
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