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A317658
Number of positions in the n-th free pure symmetric multifunction (with empty expressions allowed) with one atom.
16
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
1,2
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.
LINKS
Mathematica Reference, Orderless
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].
EXAMPLE
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[][][][][][][]
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}]
CROSSREFS
First differs from A277615 at a(128) = 5, A277615(128) = 6.
Sequence in context: A359034 A124056 A030396 * A277615 A064066 A297025
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 03 2018
STATUS
approved