A348086 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).

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, 13, 39, 36, 27, 42, 32, 34, 17, 51, 45, 40, 38, 19, 57, 48, 44, 46, 23, 69, 54, 50, 55, 60, 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

Offset: 1

Author

Scott R. Shannon, Sep 28 2021

Keywords

• nonn

Comments

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.

The fixed points below 10000 are 1, 2, 8, 10, 14, 26, 32, 38, 259. It is likely no more exist.

Examples

			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.
		

Crossrefs

Cf. A064413, A000005, A347984, A098550, A336957.

Programs

Mathematica

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 *)