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A265576 LCM-transform of EKG sequence A064413.

%I A265576 #34 Apr 22 2025 15:26:13
%S A265576 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 A265576 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 A265576 37,1,1,1,1,1,1,41,1,1,3,1,1,1,43,1,1,1,1,1,1,1,47,1,1,1,1,1,1,1,1,1,1,53
%N A265576 LCM-transform of EKG sequence A064413.
%C A265576 This is not equal to A383293(n) = 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 A265576 Antti Karttunen, <a href="/A265576/b265576.txt">Table of n, a(n) for n = 1..20000</a>
%H A265576 Andrzej Nowicki, <a href="http://arxiv.org/abs/1310.2416">Strong divisibility and LCM-sequences</a>, arXiv:1310.2416 [math.NT], 2013.
%H A265576 Andrzej 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.
%H A265576 <a href="/index/Ed#EKG">Index entries for sequences related to EKG sequence</a>.
%F A265576 a(n) = lcm {1..A064413(n)} / lcm {1..A064413(n-1)}. - _Antti Karttunen_, Apr 21 2025
%p A265576 LCMXfm:=proc(a) local L,i,n,g,b;
%p A265576 L:=nops(a);
%p A265576 g:=Array(1..L,0); b:=Array(1..L,0);
%p A265576 b[1]:=a[1]; g[1]:=a[1];
%p A265576 for n from 2 to L do g[n]:=ilcm(g[n-1],a[n]); b[n]:=g[n]/g[n-1]; od;
%p A265576 lprint([seq(b[i],i=1..L)]);
%p A265576 end;
%p A265576 # let t1 contain the first 100 terms of A064413
%p A265576 LCMXfm(t1);
%t A265576 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 A265576 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 A265576 LCMXfm[Array[ekg, 100]] (* _Jean-François Alcover_, Dec 05 2017 *)
%o A265576 (PARI)
%o A265576 LCMtransform(v) = { my(len = length(v), b = vector(len), g = vector(len)); b[1] = g[1] = 1; for(n=2, len, g[n] = lcm(g[n-1], v[n]); b[n] = g[n]/g[n-1]); (b); };
%o A265576 up_to = 20000;
%o A265576 v265576 = LCMtransform(vector(up_to, i, A064413(i))); \\ With precomputed A064413.
%o A265576 A265576(n) = v265576[n]; \\ _Antti Karttunen_, Apr 21 2025
%Y A265576 Cf. A064413, A383284 (rgs-transform), A383285 (positions of terms > 1), A383295.
%Y A265576 Positions of records: {2} U A064423.
%Y A265576 Other LCM-transforms are A014963, A061446, A265574, A265575, A368900 (see the last one for many other examples), A383258.
%Y A265576 Cf. also A383293.
%K A265576 nonn
%O A265576 1,2
%A A265576 _N. J. A. Sloane_, Jan 02 2016
%E A265576 More terms from _Antti Karttunen_, Apr 21 2025