Revision History for A317658
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Showing entries 1-10
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#11 by Alois P. Heinz at Tue Sep 11 21:15:38 EDT 2018
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#10 by Gus Wiseman at Tue Sep 11 18:22:51 EDT 2018
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#9 by Gus Wiseman at Tue Sep 11 18:22:06 EDT 2018
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| COMMENTS
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Also the number of positions in the orderless Mathematica expression with e-number n.
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| EXAMPLE
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The first twenty free pure symmetric multifunction (with emptyMathematica expressions allowed)::
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Discussion
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| Tue Sep 11
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| Gus Wiseman: Removed "Mathematica" from the name.
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#8 by Gus Wiseman at Tue Sep 11 01:51:05 EDT 2018
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| NAME
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Number of positions in the n-th free pure symmetric orderlessmultifunction (with Mathematicaempty expressions allowed) with one atom.
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| COMMENTS
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The definition ofGiven orderlessa Mathematicapositive expressioninteger isn > 1 similarwe toconstruct thata ofunique free pure symmetric multifunction (see A280000e(n) exceptby expressing n as a power of a number that is not a perfect power to a product of emptyprime expressionsnumbers: n = rad(x)^(prime(y_1) * ... * prime(y_k)) where f[] arerad = A007916. Then allowed.e(n) = e(x)[e(y_1), ..., e(y_k)].
Given a positive integer n > 1 we construct a unique orderless Mathematica expression 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)].
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| EXAMPLE
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The first twenty free pure symmetric orderlessmultifunction (with Mathematicaempty expressions: allowed):
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approved
editing
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#7 by Susanna Cuyler at Fri Aug 03 08:17:20 EDT 2018
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#6 by Gus Wiseman at Fri Aug 03 06:02:47 EDT 2018
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#5 by Gus Wiseman at Fri Aug 03 06:01:46 EDT 2018
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| FORMULA
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e(2^prime(2^prime(2^^n) = ^...))) = o[o[...o[xo]]].
e(primerad(prime(...primerad(1)))) = rad(...)^2)^2)^2) = o[o][o]...[o].
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#4 by Gus Wiseman at Fri Aug 03 05:42:45 EDT 2018
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| COMMENTS
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Given a positive integer n > 1 we construct a unique orderless Mathematica expression 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)].
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#3 by Gus Wiseman at Fri Aug 03 04:56:14 EDT 2018
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| CROSSREFS
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Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000, A317652, A317653, A317654, A317655, A317656.
Cf. A317652, A317653, A317654, A317655, A317656.
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#2 by Gus Wiseman at Fri Aug 03 04:49:25 EDT 2018
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| NAME
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allocatedNumber of positions in the n-th orderless Mathematica expressions forwith Gusone Wisemanatom.
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| DATA
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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
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| OFFSET
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1,2
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| COMMENTS
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The definition of orderless Mathematica expression is similar to that of free pure symmetric multifunction (see A280000) except that empty expressions f[] are allowed.
Given a positive integer n > 1 we construct a unique orderless Mathematica expression 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)].
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| LINKS
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Mathematica Reference, <a href="http://reference.wolfram.com/mathematica/ref/Orderless.html">Orderless</a>
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| FORMULA
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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^^n) = o[o[...o[x]]].
e(prime(prime(...prime(1)))) = o[o][o]...[o].
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| EXAMPLE
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The first twenty orderless 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[][][][][][][]
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| MATHEMATICA
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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}]
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| CROSSREFS
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First differs from A277615 at a(128) = 5, A277615(128) = 6.
Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000, A317652, A317653, A317654, A317655, A317656.
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| KEYWORD
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allocated
nonn
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| AUTHOR
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Gus Wiseman, Aug 03 2018
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| STATUS
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approved
editing
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