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%I #14 Mar 20 2022 12:40:14
%S 1,2,4,6,3,9,12,8,16,10,5,15,18,14,7,21,24,20,22,11,33,30,25,35,28,26,
%T 13,39,36,27,42,32,34,17,51,45,40,38,19,57,48,44,46,23,69,54,50,55,60,
%U 52,56,49,63,66,58,29,87,72,62,31,93,75,65,70,64,68,74,37,111,78,76,80,82,41,123,81
%N Variation of the EKG sequence A064413: a(n) is the smallest number not already used which shares a factor with a(n-1) and has a different number of divisors than a(n-1).
%C This sequence uses the same rules as A064413 except with the additional restriction that a(n) must have a different number of divisors than a(n-1). Despite this additional restriction the overall behavior of the sequence is almost identical to A064413, with most values appearing only a few places away from their position in A064413.
%C The fixed points below 10000 are 1, 2, 8, 10, 14, 26, 32, 38, 259. It is likely no more exist.
%e a(9) = 16 as a(8) = 8 which shares a factor with 16, 16 is unused, and tau(8) = 4 does not equal tau(16) = 5. This is the first term that differs from A064413, the later having a(9) = 10, but tau(10) = 4 = tau(8) so is forbidden here.
%t a[1]=1;a[2]=2;a[n_]:=a[n]=(k=1;While[GCD[k,a[n-1]]==1||SameQ@@DivisorSigma[0,{k,a[n-1]}]||MemberQ[Array[a,n-1],k],k++];k);Array[a,76] (* _Giorgos Kalogeropoulos_, Oct 08 2021 *)
%Y Cf. A064413, A000005, A347984, A098550, A336957.
%K nonn
%O 1,2
%A _Scott R. Shannon_, Sep 28 2021