cp's OEIS Frontend

A349323 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.

%I A349323 #35 Jul 04 2025 14:30:25
%S A349323 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 A349323 1296,17,324,19,169,21,196,22,225,23,256,24,289,26,361,27,400,29,441,
%U A349323 30,484,31,529,33,576,34,64,35,729,18,5184,20,2401,28,10000,32,11664
%N A349323 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 A349323 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).
%H A349323 Michael De Vlieger, <a href="/A349323/b349323.txt">Table of n, a(n) for n = 1..10000</a>
%H A349323 Michael De Vlieger, <a href="/A349323/a349323_1.png">Log log scatterplot of a(n)</a>, n = 1..10000, with a color function showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and magenta, where magenta represents powerful numbers that are not prime powers. Squares comprise the upper trajectory, whereas the lower trajectory contains relatively few squares.
%e A349323 a(1)=1, a(2)=2, a(3)=4, with number of divisors 1,2,3 respectively.
%e A349323 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 A349323 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 A349323 (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 A349323 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 A349323 Cf. A350150, A098550, A000005, A000290, A000037.
%K A349323 nonn
%O A349323 1,2
%A A349323 _David James Sycamore_, Dec 22 2021
%E A349323 More terms from _Michael De Vlieger_, Dec 24 2021