# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: id:a317658
Showing 1-1 of 1

%I A317658 #11 Sep 11 2018 21:15:38
%S A317658 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,
%T A317658 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,
%U A317658 5,9,8,10,12,12,8,13,9,10,14,6,10,9,11,13,13
%N A317658 Number of positions in the n-th free pure symmetric multifunction (with empty expressions allowed) with one atom.
%C A317658 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)].
%C A317658 Also the number of positions in the orderless Mathematica expression with e-number n.
%H A317658 Mathematica Reference, <a href="http://reference.wolfram.com/mathematica/ref/Orderless.html">Orderless</a>
%F A317658 a(rad(x)^(prime(y_1) * ... * prime(y_k)) = a(x) + a(y_1) + ... + a(y_k).
%F A317658 e(2^(2^n)) = o[o,...,o].
%F A317658 e(2^prime(2^prime(2^...))) = o[o[...o[o]]].
%F A317658 e(rad(rad(rad(...)^2)^2)^2) = o[o][o]...[o].
%e A317658 The first twenty Mathematica expressions:
%e A317658    1: o
%e A317658    2: o[]
%e A317658    3: o[][]
%e A317658    4: o[o]
%e A317658    5: o[][][]
%e A317658    6: o[o][]
%e A317658    7: o[][][][]
%e A317658    8: o[o[]]
%e A317658    9: o[][o]
%e A317658   10: o[o][][]
%e A317658   11: o[][][][][]
%e A317658   12: o[o[]][]
%e A317658   13: o[][o][]
%e A317658   14: o[o][][][]
%e A317658   15: o[][][][][][]
%e A317658   16: o[o,o]
%e A317658   17: o[o[]][][]
%e A317658   18: o[][o][][]
%e A317658   19: o[o][][][][]
%e A317658   20: o[][][][][][][]
%t A317658 nn=100;
%t A317658 radQ[n_]:=If[n===1,False,GCD@@FactorInteger[n][[All,2]]===1];
%t A317658 rad[n_]:=rad[n]=If[n===0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];
%t A317658 Clear[radPi];Set@@@Array[radPi[rad[#]]==#&,nn];
%t A317658 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]]]]]];
%t A317658 Table[exp[n],{n,1,nn}]
%Y A317658 First differs from A277615 at a(128) = 5, A277615(128) = 6.
%Y A317658 Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000.
%Y A317658 Cf. A317652, A317653, A317654, A317655, A317656.
%K A317658 nonn
%O A317658 1,2
%A A317658 _Gus Wiseman_, Aug 03 2018

# Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE