A317658 Number of positions in the n-th free pure symmetric multifunction (with empty expressions allowed) with one atom.

1, 2, 3, 3, 4, 4, 5, 4, 4, 5, 6, 5, 5, 6, 7, 4, 6, 6, 7, 8, 5, 7, 7, 8, 5, 9, 5, 6, 8, 8, 9, 5, 6, 10, 6, 5, 7, 9, 9, 10, 6, 7, 11, 7, 6, 8, 10, 10, 6, 11, 7, 8, 12, 8, 7, 9, 11, 11, 7, 12, 8, 9, 13, 5, 9, 8, 10, 12, 12, 8, 13, 9, 10, 14, 6, 10, 9, 11, 13, 13

Offset: 1

Author

Gus Wiseman, Aug 03 2018

Keywords

• nonn

Comments

Given a positive integer n > 1 we construct a unique free pure symmetric multifunction e(n) by expressing n as a power of a number that is not a perfect power to a product of prime numbers: n = rad(x)^(prime(y_1) * ... * prime(y_k)) where rad = A007916. Then e(n) = e(x)[e(y_1), ..., e(y_k)].

Also the number of positions in the orderless Mathematica expression with e-number n.

Examples

			The first twenty Mathematica expressions:
   1: o
   2: o[]
   3: o[][]
   4: o[o]
   5: o[][][]
   6: o[o][]
   7: o[][][][]
   8: o[o[]]
   9: o[][o]
  10: o[o][][]
  11: o[][][][][]
  12: o[o[]][]
  13: o[][o][]
  14: o[o][][][]
  15: o[][][][][][]
  16: o[o,o]
  17: o[o[]][][]
  18: o[][o][][]
  19: o[o][][][][]
  20: o[][][][][][][]
		

Links

• Mathematica Reference, Orderless

Crossrefs

First differs from A277615 at a(128) = 5, A277615(128) = 6.

Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000.

Cf. A317652, A317653, A317654, A317655, A317656.

Programs

Mathematica

nn=100;
radQ[n_]:=If[n===1,False,GCD@@FactorInteger[n][[All,2]]===1];
rad[n_]:=rad[n]=If[n===0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];
Clear[radPi];Set@@@Array[radPi[rad[#]]==#&,nn];
exp[n_]:=If[n===1,x,With[{g=GCD@@FactorInteger[n][[All,2]]},Apply[exp[radPi[Power[n,1/g]]],exp/@Flatten[Cases[FactorInteger[g],{p_?PrimeQ,k_}:>ConstantArray[PrimePi[p],k]]]]]];
Table[exp[n],{n,1,nn}]

Formula

a(rad(x)^(prime(y_1) * ... * prime(y_k))) = a(x) + a(y_1) + ... + a(y_k).

e(2^(2^n)) = o[o,...,o].

e(2^prime(2^prime(2^...))) = o[o[...o[o]]].

e(rad(rad(rad(...)^2)^2)^2) = o[o][o]...[o].