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A279944 Number of positions in the free pure symmetric multifunction in one symbol with j-number 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 (list; graph; refs; listen; history; text; internal format)
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).

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[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

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 A087243

Adjacent sequences:  A279941 A279942 A279943 * A279945 A279946 A279947

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 24 2016

STATUS

approved

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Last modified February 19 10:03 EST 2020. Contains 332041 sequences. (Running on oeis4.)