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a(1)=1, a(2)=2, a(3)=4. Thereafter, for n>=3, a(n+1) is the smallest unused k such that d(k) is prime to both d(a(n)) and d(a(n-2)), but not to d(a(n-1)), where d is the divisor counting (tau) function A000005.
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%I #29 Feb 11 2022 16:58:47

%S 1,2,4,3,9,5,25,6,36,7,49,8,100,10,121,11,144,13,16,14,81,12,625,15,

%T 1296,17,324,19,169,21,196,22,225,23,256,24,289,26,361,27,400,29,441,

%U 30,484,31,529,33,576,34,64,35,729,18,5184,20,2401,28,10000,32,11664

%N a(1)=1, a(2)=2, a(3)=4. Thereafter, for n>=3, a(n+1) is the smallest unused k such that d(k) is prime to both d(a(n)) and d(a(n-2)), but not to d(a(n-1)), where d is the divisor counting (tau) function A000005.

%C Permutation of the positive integers, a "Yellowstone" version of A350150, having similar characteristics to the latter. The sequence interleaves squares a(2n+1) having odd tau with nonsquares a(2n) having even tau. Numbers with the same tau appear in their natural order (primes, squares, etc).

%e a(1)=1, a(2)=2, a(3)=4, with number of divisors 1,2,3 respectively.

%e a(4) must be 3 because d(3)=2, which is prime to d(a(3))=d(4)=3 and to d(a(1))=d(1)=1 but it is not prime to d(a(2))=d(2)=2, and 3 is the least unused number with this property.

%t Nest[Block[{a = #1, i = #2, j = #3, k = #4, m = 3}, While[Nand[FreeQ[a, m], CoprimeQ[#, i], ! CoprimeQ[#, j], CoprimeQ[#, k]] &@DivisorSigma[0, m], m++]; Append[#1, m]] & @@ Join[{#}, DivisorSigma[0, #[[-3 ;; -1]]]] &, {1, 2, 4}, 58] (* _Michael De Vlieger_, Jan 15 2022 *)

%o (PARI) isok(k, ndx, ndy, ndz, set) = {if (!setsearch(set, k), my(ndk=numdiv(k)); (gcd(ndx,ndk)==1) && (gcd(ndy,ndk)!=1) && (gcd(ndz,ndk)==1););}

%o lista(nn) = {my(x=1, y=2, z=4, list=List([x,y,z]), set = Set(list)); for (n=4, nn, my(k=1, ndx=numdiv(x), ndy=numdiv(y), ndz=numdiv(z)); while (!isok(k, ndx, ndy, ndz, set), k++); listput(list, k); set = Set(list); x=y; y=z; z=k;); Vec(list);} \\ _Michel Marcus_, Jan 16 2022

%Y Cf. A350150, A098550, A000005, A000290 (squares), A000037 (nonsquares).

%K nonn

%O 1,2

%A _David James Sycamore_, Dec 22 2021

%E More terms from _Michael De Vlieger_, Dec 24 2021