A394540 Sequence where, for every step t starting from 2, k is appended if t is a multiple of s(k-1), where s is Sylvester's sequence (A000058). If multiple numbers trigger on the same step, they are appended in increasing order.

1, 2, 1, 1, 2, 3, 1, 2, 1, 1, 2, 1, 3, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 1, 2, 3, 4, 1, 2, 1, 1, 2, 3, 1, 2, 1, 1, 2, 1, 3, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 1, 2, 3, 1, 4, 2, 1

Offset

1,2

Comments

The density of k in this sequence is 1/s(k-1). This can be viewed as a probability mass function since Sum_{k>0} 1/s(k-1) = 1.

The arithmetic mean approaches Sum_{k>0} k/s(k-1) = 1.69103020... in the limit.

The geometric mean approaches Product_{k>1} k^(1/s(k-1)) = 1.523673650289... in the limit.

The sequence is generated by the D'Hondt (or Jefferson) apportionment method applied to the probability mass function above. - Pontus von Brömssen, Apr 20 2026

Links

Jwalin Bhatt, Table of n, a(n) for n = 1..10000

Wikipedia, D'Hondt method.

Wikipedia, Sylvester's sequence.

Example

At step t=2, we append a 1 because 2 is a multiple of s(0)=2.
At step t=3, we append a 2 because 3 is a multiple of s(1)=3.
At step t=4, we append a 1 because 4 is a multiple of s(0)=2.
At step t=6, we append a 1 and then 2 because 6 is a multiple of both 2 and 3.
At step t=7, we append a 3 because 7 is a multiple of s(2)=7.

Mathematica

seq[len_] := Module[{n = 1, m = 1, c = 0, v = {}, s}, s[0] = 2; s[k_] := s[k] = s[k - 1]^2 - s[k - 1] + 1; While[c < len, n++; Do[If[Divisible[n, s[i - 1]], AppendTo[v, i]; c++; If[i == m + 1, m++]], {i, m + 1}]]; v]; seq[100] (* Amiram Eldar, Mar 24 2026 *)

Prog

(Python)
from itertools import islice
def sylvester(k):
    if k == 0: return 2
    s = sylvester(k-1)
    return s*s - s + 1
def sample_sylvester_distribution():
    n, num_reached = 1, 1
    while n := n+1:
        for num in range(1, num_reached+2):
            if n % sylvester(num-1) == 0:
                yield num
                num_reached += num == num_reached+1
A394540 = list(islice(sample_sylvester_distribution(), 120))

Crossrefs

Cf. A000058, A381522, A381900, A382093, A394854.

Sequence in context: A184219 A180262 A395698 * A161789 A109671 A141289

Adjacent sequences: A394537 A394538 A394539 * A394541 A394542 A394543

Keyword

nonn

Author

Jwalin Bhatt, Mar 24 2026

Status

approved