

A279944


Number of positions in the free pure symmetric multifunction in one symbol with jnumber n.


15



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

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 nth number that is not a perfect power (A007916) and p(n) is the nth prime number (A000040). See example. Then a(n) is the number of brackets [...] plus the number of x's in j(n).


LINKS

Table of n, a(n) for n=1..100.


FORMULA

a(A007916(h)^(A000040(g_1)*...*A000040(g_k)1)) = 1 + a(h) + a(g_1) + ... + a(g_k).


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[n1]+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[n1]]];
Array[diwt, nn]


CROSSREFS

Cf. A005043, A007916, A106490, A277564, A277615, A277996, A278028, A280000.
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]]]).
Sequence in context: A307701 A103332 A195796 * A309424 A079886 A338363
Adjacent sequences: A279941 A279942 A279943 * A279945 A279946 A279947


KEYWORD

nonn


AUTHOR

Gus Wiseman, Dec 24 2016


STATUS

approved



