A265576 LCM-transform of EKG sequence A064413.

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, 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, 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

Offset: 1

Author

N. J. A. Sloane, Jan 02 2016

Keywords

• nonn

Comments

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

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Andrzej Nowicki, Strong divisibility and LCM-sequences, arXiv:1310.2416 [math.NT], 2013.
• Andrzej Nowicki, Strong divisibility and LCM-sequences, Am. Math. Mnthly 122 (2015), 958-966.
• Index entries for sequences related to EKG sequence.

Crossrefs

Cf. A064413, A383284 (rgs-transform), A383285 (positions of terms > 1), A383295.

Positions of records: {2} U A064423.

Other LCM-transforms are A014963, A061446, A265574, A265575, A368900 (see the last one for many other examples), A383258.

Cf. also A383293.

Programs

Maple

LCMXfm:=proc(a) local L,i,n,g,b;
L:=nops(a);
g:=Array(1..L,0); b:=Array(1..L,0);
b[1]:=a[1]; g[1]:=a[1];
for n from 2 to L do g[n]:=ilcm(g[n-1],a[n]); b[n]:=g[n]/g[n-1]; od;
lprint([seq(b[i],i=1..L)]);
end;
# let t1 contain the first 100 terms of A064413
LCMXfm(t1);

Mathematica

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]];
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]]];
LCMXfm[Array[ekg, 100]] (* Jean-François Alcover, Dec 05 2017 *)

PARI

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); };
up_to = 20000;
v265576 = LCMtransform(vector(up_to, i, A064413(i))); \\ With precomputed A064413.
A265576(n) = v265576[n]; \\ Antti Karttunen, Apr 21 2025

Formula

a(n) = lcm {1..A064413(n)} / lcm {1..A064413(n-1)}. - Antti Karttunen, Apr 21 2025

Extensions

More terms from Antti Karttunen, Apr 21 2025