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A354903
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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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2
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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PROG
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(Python)
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
(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
(C) See Links section.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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