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

%I A354903 #31 Jul 11 2025 18:40:20
%S A354903 1,2,4,9,3,5,6,16,25,7,8,36,64,49,10,100,121,11,12,81,13,14,144,625,
%T A354903 15,17,18,729,19,20,169,21,22,196,225,23,24,1024,256,289,26,324,1296,
%U A354903 2401,27,29,28,361,30,4096,400,441,31,32,484,529,33,34,576,5184
%N A354903 Lexicographically earliest infinite sequence of distinct positive integers such that the number of divisors of a(n+1) is prime to a(n).
%C A354903 1,2 are the earliest consecutive pair of numbers satisfying the definition, therefore the sequence begins with a(1)=1, a(2)=2.
%C A354903 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.
%C A354903 All record terms are squares, though not in ascending order (64 occurs before 49, 100 before 81, etc.).
%C A354903 Conjectured to be a permutation of the positive integers in which primes appear in natural order.
%H A354903 Michael De Vlieger, <a href="/A354903/b354903.txt">Table of n, a(n) for n = 1..16384</a> first 1237 terms from Rémy Sigrist.
%H A354903 Michael De Vlieger, <a href="/A354903/a354903_1.txt">Mathematica code</a>.
%H A354903 Rémy Sigrist, <a href="/A354903/a354903.txt">C program</a>
%e A354903 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.
%o A354903 (Python)
%o A354903 from math import gcd
%o A354903 from sympy import divisor_count
%o A354903 from itertools import count, islice
%o A354903 def agen(): # generator of terms
%o A354903     aset, k, mink = {1}, 1, 2; yield 1
%o A354903     for n in count(2):
%o A354903         an, k = k, mink
%o A354903         while k in aset or not gcd(an, divisor_count(k)) == 1: k += 1
%o A354903         aset.add(k); yield k
%o A354903         while mink in aset: mink += 1
%o A354903 print(list(islice(agen(), 60))) # _Michael S. Branicky_, Jun 11 2022
%o A354903 (PARI) 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
%o A354903 (C) // See Links section.
%Y A354903 Cf. A000005, A005179, A000290, A350150, A355269.
%K A354903 nonn
%O A354903 1,2
%A A354903 _David James Sycamore_, Jun 11 2022
%E A354903 a(15) and beyond from _Michael S. Branicky_, Jun 11 2022