Revision History for A292127
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing all changes.
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#10 by Susanna Cuyler at Fri May 04 22:42:29 EDT 2018
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#9 by Jon E. Schoenfield at Thu May 03 22:02:48 EDT 2018
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#8 by Jon E. Schoenfield at Thu May 03 22:02:46 EDT 2018
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| NAME
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a(1) = 1, a(r(n)^k) = 1 + k * a(n) where r(n) is the n-th rootless number that is not a perfect power A007916(n).
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| STATUS
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approved
editing
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#7 by N. J. A. Sloane at Sun Oct 01 00:06:12 EDT 2017
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#6 by Gus Wiseman at Mon Sep 11 00:59:01 EDT 2017
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#5 by Gus Wiseman at Mon Sep 11 00:55:21 EDT 2017
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| MATHEMATICA
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radQrads=Select[n_]:=IfRange[n===1, False2, nn], GCD@@FactorInteger[n][[[#][[All, 2]]===1]; &];
rada[1]:=1; a[n_]:=rad[n]=IfWith[{k=GCD@@FactorInteger[n===0, 1][[All, NestWhile[#+2]]}, 1&, rad+k*a[Position[rads, n-^(1]+/k)][[1, Not[radQ[#]]&]]; 1]]]];
Clear[radPi]; Set@@@Array[radPi[rad[#]]==#&, nn];
tct[n_]:=If[n===1, 1, With[{g=GCD@@FactorInteger[n][[All, 2]]}, ConstantArray[tct[radPi[Power[n, 1/g]]], g]]];
Table[Count[tct[n], _, {0, Infinity}], {n, nn}]
Array[a, nn]
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#4 by Gus Wiseman at Sat Sep 09 04:29:06 EDT 2017
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| NAME
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a(1) = 1, a(r(n)^k) = 1+ + k* * a(n) where r(n) is the n-th rootless number A007916(n).
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| COMMENTS
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Any positive integer greater than 1 can be written uniquely as a perfect power r(n)^k. We define a planted achiral (or generalized Bethe) tree b(n) for any positive integer greater than 1 by writing n as a perfect power r(d)^k and forming a tree with k branches all equal to b(d). Then a(n) is the number of nodes in b(n).
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#3 by Gus Wiseman at Sat Sep 09 03:28:07 EDT 2017
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| EXAMPLE
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The first twentynineteen planted achiral trees are:
((((((o)))))), (oooo), (((ooo))), ((((o)(o)))), (((((oo))))), (((((((o))))))).))))).
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#2 by Gus Wiseman at Sat Sep 09 03:05:45 EDT 2017
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| NAME
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allocateda(r(n)^k) = 1+k*a(n) where r(n) is the n-th forrootless Gusnumber WisemanA007916(n).
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| DATA
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1, 2, 3, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6, 6, 7, 5, 6, 7, 7, 8, 6, 7, 8, 8, 7, 9, 7, 7, 8, 9, 9, 6, 8, 10, 8, 7, 8, 9, 10, 10, 7, 9, 11, 9, 8, 9, 10, 11, 9, 11, 8, 10, 12, 10, 9, 10, 11, 12, 10, 12, 9, 11, 13, 7, 11, 10, 11, 12, 13, 11, 13, 10, 12, 14, 8, 12, 11
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| OFFSET
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1,2
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| COMMENTS
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Any positive integer can be written uniquely as a perfect power r(n)^k. We define a planted achiral (or generalized Bethe) tree b(n) for any positive integer by writing n as a perfect power r(d)^k and forming a tree with k branches all equal to b(d). Then a(n) is the number of nodes in b(n).
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| EXAMPLE
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The first twenty planted achiral trees are:
o,
(o),
((o)), (oo),
(((o))), ((oo)),
((((o)))), (ooo), ((o)(o)), (((oo))),
(((((o))))), ((ooo)), (((o)(o))), ((((oo)))),
((((((o)))))), (oooo), (((ooo))), ((((o)(o)))), (((((oo))))), (((((((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];
tct[n_]:=If[n===1, 1, With[{g=GCD@@FactorInteger[n][[All, 2]]}, ConstantArray[tct[radPi[Power[n, 1/g]]], g]]];
Table[Count[tct[n], _, {0, Infinity}], {n, nn}]
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| CROSSREFS
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Cf. A003238, A007916, A052409, A052410, A061775, A214577, A277576, A277615, A278028, A279614, A279944, A289023.
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| KEYWORD
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allocated
nonn
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| AUTHOR
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Gus Wiseman, Sep 09 2017
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| STATUS
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approved
editing
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#1 by Gus Wiseman at Sat Sep 09 03:05:45 EDT 2017
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| NAME
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allocated for Gus Wiseman
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| KEYWORD
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allocated
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| STATUS
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approved
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