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LCM-transform of Euler totient numbers (A000010).
4

%I #19 Nov 09 2018 21:59:09

%S 1,1,2,1,2,1,3,1,1,1,5,1,1,1,2,1,2,1,3,1,1,1,11,1,1,1,1,1,7,1,1,1,1,1,

%T 1,1,1,1,1,1,1,1,1,1,1,1,23,1,1,1,2,1,13,1,1,1,1,1,29,1,1,1,1,1,1,1,1,

%U 1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,41,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1

%N LCM-transform of Euler totient numbers (A000010).

%H Antti Karttunen, <a href="/A265575/b265575.txt">Table of n, a(n) for n = 1..26003</a>

%H Antti Karttunen, <a href="/A265575/a265575.txt">Data supplement: n, a(n) computed for n = 1..100000</a>

%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 with(numtheory);

%p t1:=[seq(phi(n),n=1..100)];

%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 LCMXfm[Table[EulerPhi[n], {n, 1, 100}]] (* _Jean-François Alcover_, Dec 05 2017, from Maple *)

%o (PARI)

%o up_to = 10000;

%o 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 v265575 = LCMtransform(vector(up_to,i,eulerphi(i)));

%o A265575(n) = v265575[n]; \\ _Antti Karttunen_, Nov 09 2018

%Y Cf. A000010.

%Y Other LCM-transforms are A061446, A265574, A265576, A265577, A265578.

%K nonn

%O 1,3

%A _N. J. A. Sloane_, Jan 02 2016