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

1,2

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)

Links

Yifan Xie, Table of n, a(n) for n = 1..10000

Index entries for sequences that are permutations of the natural numbers

Example

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

Prog

(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

Crossrefs

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

Sequence in context: A201281 A095303 A060734 * A075594 A076022 A064421

Adjacent sequences: A375716 A375717 A375718 * A375720 A375721 A375722

Keyword

nonn,easy

Author

Yifan Xie, Aug 25 2024

Status

approved