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A265578 LCM-transform of number of divisors function (A000005).

%I A265578 #19 Nov 06 2018 21:18:55
%S A265578 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 A265578 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 A265578 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 A265578 LCM-transform of number of divisors function (A000005).
%C A265578 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 A265578 Antti Karttunen, <a href="/A265578/b265578.txt">Table of n, a(n) for n = 1..16384</a>
%H A265578 A. Nowicki, <a href="http://arxiv.org/abs/1310.2416">Strong divisibility and LCM-sequences</a>, arXiv:1310.2416 [math.NT], 2013.
%H A265578 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 A265578 Antti Karttunen, <a href="/A265578/a265578.txt">Data supplement: n, a(n) computed for n =  1..100000</a>
%p A265578 LCMXfm:=proc(a) local L,i,n,g,b;
%p A265578 L:=nops(a);
%p A265578 g:=Array(1..L,0); b:=Array(1..L,0);
%p A265578 b[1]:=a[1]; g[1]:=a[1];
%p A265578 for n from 2 to L do g[n]:=ilcm(g[n-1],a[n]); b[n]:=g[n]/g[n-1]; od;
%p A265578 lprint([seq(b[i],i=1..L)]);
%p A265578 end;
%p A265578 with(numtheory);
%p A265578 t1:=[seq(tau(n),n=1..100)];
%p A265578 LCMXfm(t1);
%t A265578 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 A265578 LCMXfm[Table[DivisorSigma[0, n], {n, 1, 100}]] (* _Jean-François Alcover_, Dec 05 2017, from Maple *)
%o A265578 (PARI)
%o A265578 up_to = 16384;
%o A265578 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 A265578 v265578 = LCMtransform(vector(up_to,i,numdiv(i)));
%o A265578 A265578(n) = v265578[n]; \\ _Antti Karttunen_, Nov 06 2018
%Y A265578 Cf. A000005.
%Y A265578 Other LCM-transforms are A061446, A265574, A265575, A265576, A265577.
%K A265578 nonn
%O A265578 1,2
%A A265578 _N. J. A. Sloane_, Jan 02 2016