|
%I #21 Jan 13 2024 10:43:12
%S 1,2,2,3,1,3,1,2,5,1,1,1,7,1,1,1,2,1,11,1,1,3,1,5,1,1,13,1,1,1,2,17,1,
%T 1,1,19,1,1,1,1,1,23,1,1,1,1,1,1,1,7,1,1,1,1,1,29,1,1,1,31,1,1,1,2,1,
%U 37,1,1,1,1,1,1,41
%N LCM-transform of EKG sequence A064413.
%C This is not equal to A014963(A064413(n)) because the EKG-permutation doesn't satisfy the property that all prime powers should appear before any of their multiples, as, for example, A064413(4) = 6 comes before A064413(5) = 3. See comments in A368900. - _Antti Karttunen_, Jan 13 2024
%H A. Nowicki, <a href="http://arxiv.org/abs/1310.2416">Strong divisibility and LCM-sequences</a>, arXiv:1310.2416 [math.NT], 2013.
%H A. Nowicki, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.122.10.958">Strong divisibility and LCM-sequences</a>, Am. Math. Mnthly 122 (2015), 958-966.
%p LCMXfm:=proc(a) local L,i,n,g,b;
%p L:=nops(a);
%p g:=Array(1..L,0); b:=Array(1..L,0);
%p b[1]:=a[1]; g[1]:=a[1];
%p for n from 2 to L do g[n]:=ilcm(g[n-1],a[n]); b[n]:=g[n]/g[n-1]; od;
%p lprint([seq(b[i],i=1..L)]);
%p end;
%p # let t1 contain the first 100 terms of A064413
%p LCMXfm(t1);
%t LCMXfm[a_List] := Module[{L = Length[a], b, g}, b[1] = g[1] = a[[1]]; b[_] = 0; g[_] = 0; Do[g[n] = LCM[g[n - 1], a[[n]]]; b[n] = g[n]/g[n - 1], {n, 2, L}]; Array[b, L]];
%t ekg[1] = 1; ekg[2] = 2; ekg[n_] := ekg[n] = For[k = 1, True, k++, If[FreeQ[ Array[ekg, n - 1], k] && !CoprimeQ[k, ekg[n - 1]], Return[k]]];
%t LCMXfm[Array[ekg, 100]] (* _Jean-François Alcover_, Dec 05 2017 *)
%Y Cf. A064413.
%Y Other LCM-transforms are A014963, A061446, A265574, A265575, A368900 (see the last one for many other examples).
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Jan 02 2016
|
