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.

1, 2, 4, 3, 9, 5, 25, 6, 36, 7, 49, 8, 100, 10, 121, 11, 144, 13, 16, 14, 81, 12, 625, 15, 1296, 17, 324, 19, 169, 21, 196, 22, 225, 23, 256, 24, 289, 26, 361, 27, 400, 29, 441, 30, 484, 31, 529, 33, 576, 34, 64, 35, 729, 18, 5184, 20, 2401, 28, 10000, 32, 11664

Offset

1,2

Comments

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).

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Log log scatterplot of a(n), 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.

Example

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

Mathematica

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 *)

Prog

(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); ); }
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

Crossrefs

Cf. A350150, A098550, A000005, A000290, A000037.

Sequence in context: A137442 A350150 A111390 * A344554 A129596 A356222

Adjacent sequences: A349320 A349321 A349322 * A349324 A349325 A349326

Keyword

nonn

Author

David James Sycamore, Dec 22 2021

Extensions

More terms from Michael De Vlieger, Dec 24 2021

Status

approved