A350150 -id:A350150 - cp's OEIS frontend

A354903 Lexicographically earliest infinite sequence of distinct positive integers such that the number of divisors of a(n+1) is prime to a(n).

1, 2, 4, 9, 3, 5, 6, 16, 25, 7, 8, 36, 64, 49, 10, 100, 121, 11, 12, 81, 13, 14, 144, 625, 15, 17, 18, 729, 19, 20, 169, 21, 22, 196, 225, 23, 24, 1024, 256, 289, 26, 324, 1296, 2401, 27, 29, 28, 361, 30, 4096, 400, 441, 31, 32, 484, 529, 33, 34, 576, 5184

Offset: 1

David James Sycamore, Jun 11 2022

nonn

1,2 are the earliest consecutive pair of numbers satisfying the definition, therefore the sequence begins with a(1)=1, a(2)=2.
The sequence is infinite since there is always a number k prime to a(n), and the smallest number not yet used which has k divisors could be a(n+1), unless there is a smaller number with the same property.
All record terms are squares, though not in ascending order (64 occurs before 49, 100 before 81, etc.).
Conjectured to be a permutation of the positive integers in which primes appear in natural order.

			a(7)=6 and 16 is the smallest number which has not already occurred whose number of divisors (5) is prime to 6, therefore a(8)=16.

Michael De Vlieger, Table of n, a(n) for n = 1..16384 first 1237 terms from Rémy Sigrist.
Michael De Vlieger, Mathematica code.
Rémy Sigrist, C program

Cf. A000005, A005179, A000290, A350150, A355269.

C
```
// See Links section.
```

lista(nn) = my(va = vector(nn)); va[1] = 1; for (n=2, nn, my(k=1); while ((gcd(va[n-1], numdiv(k)) != 1) || #select(x->(x==k), va), k++); va[n] = k;); va; \\ Michel Marcus, Jun 11 2022

from math import gcd
from sympy import divisor_count
from itertools import count, islice
def agen(): # generator of terms
    aset, k, mink = {1}, 1, 2; yield 1
    for n in count(2):
        an, k = k, mink
        while k in aset or not gcd(an, divisor_count(k)) == 1: k += 1
        aset.add(k); yield k
        while mink in aset: mink += 1
print(list(islice(agen(), 60))) # Michael S. Branicky, Jun 11 2022

a(15) and beyond from Michael S. Branicky, Jun 11 2022

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.

Original entry on oeis.org

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

David James Sycamore, Dec 22 2021

nonn

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).

			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.

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.

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

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 *)

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

More terms from Michael De Vlieger, Dec 24 2021

cp's OEIS Frontend

A354903 Lexicographically earliest infinite sequence of distinct positive integers such that the number of divisors of a(n+1) is prime to a(n).

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