A375719 a(1) = 1; For n > 1, a(n) is the smallest number different from a(1), ..., a(n-1) such that lcm(a(1), ..., a(n)) is a perfect square.

1, 4, 2, 9, 3, 6, 12, 16, 8, 18, 24, 25, 5, 10, 15, 20, 30, 36, 40, 45, 48, 49, 7, 14, 21, 28, 35, 42, 50, 56, 60, 63, 64, 32, 70, 72, 75, 80, 81, 27, 54, 84, 90, 96, 98, 100, 105, 108, 112, 120, 121, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 126, 132, 135, 140

Offset: 1

Author

Yifan Xie, Aug 25 2024

Keywords

• nonn
• easy

Comments

This is a permutation of the positive integers.

From Yifan Xie, Jan 09 2025: (Start)

The sequence can be constructed using the following properties:

1. The squares appear in increasing order.

2. p^(2k) is immediately followed by p^(2k-1), 2*p^(2k-1), ..., (p-1)*p^(2k-1) for prime p.

3. The numbers m^2 < k < (m+1)^2 such that A006530(k) < A051119(k) appear in increasing order between m^2 and (m+1)^2.

a(n) > a(n-1) iff a(n) = p^(2k) and a(n-1) = p^(2k-1), where p is a prime. (End)

Examples

			For n = 4, a(4) is different from 1, 2, 4, and lcm(4, a(4)) is a perfect square. Therefore, a(4) = 9.
		

Links

• Yifan Xie, Table of n, a(n) for n = 1..10000
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A006530 (Gpf(n)), A051119 (largest divisor of n coprime to Gpf(n)).

Programs

PARI

seq(n)={my(b=1, a=vector(n), M=Map()); for(n=1, #a, my(k=1); while(!issquare(lcm(b,k)) || mapisdefined(M,k), k++); a[n]=k; b=lcm(b,k); mapput(M,k,1)); a} \\ Andrew Howroyd, Aug 30 2024

Python

from math import isqrt, lcm
from itertools import count, islice
def sqr(n): return isqrt(n)**2 == n
def agen(): # generator of terms
    an, aset, L, m = 1, {1}, 1, 2
    for n in count(2):
        yield an
        an = next(k for k in count(m) if k not in aset and sqr(lcm(k, L)))
        aset.add(an)
        L = lcm(L, an)
        while m in aset: m += 1
print(list(islice(agen(), 65))) # Michael S. Branicky, Aug 30 2024