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%I #19 Nov 06 2018 21:18:55
%S 1,2,1,3,1,2,1,1,1,1,1,1,1,1,1,5,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,
%T 1,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,1,1,1,1,
%U 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N LCM-transform of number of divisors function (A000005).
%C Terms larger than one occur at n = 2, 4, 6, 16, 24, 36, 64, 120, 840, 900, 1296, 7560, 44100, 46656, 83160, ... - _Antti Karttunen_, Nov 06 2018
%H Antti Karttunen, <a href="/A265578/b265578.txt">Table of n, a(n) for n = 1..16384</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.
%H Antti Karttunen, <a href="/A265578/a265578.txt">Data supplement: n, a(n) computed for n = 1..100000</a>
%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(tau(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[DivisorSigma[0, n], {n, 1, 100}]] (* _Jean-François Alcover_, Dec 05 2017, from Maple *)
%o (PARI)
%o up_to = 16384;
%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 v265578 = LCMtransform(vector(up_to,i,numdiv(i)));
%o A265578(n) = v265578[n]; \\ _Antti Karttunen_, Nov 06 2018
%Y Cf. A000005.
%Y Other LCM-transforms are A061446, A265574, A265575, A265576, A265577.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Jan 02 2016
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