A349189 -id:A349189 - cp's OEIS frontend

2, 24, 48, 240, 1440, 2400, 7440, 25920, 72000, 234000

Offset: 2

Bernard Schott, Nov 09 2021

A n-phile integer m is such that there are n positive integers b_1 < b_2 < ... < b_j < ... < b_n such that b_1 divides b_2, b_2 divides b_3, ..., b_[j-1] divides b_j, ..., b_[n-1] divides b_n, and m = b_1 + b_2 + ... + b_j + ... + b_n. A number that is not n-phile is called n-phobe.
The idea for this sequence and the words 'n-phile' and 'n-phobe' come from the French website Diophante (see link).
The number of n-phobe numbers is always finite (A349189), the smallest one is always 1, and this sequence lists the largest n-phobe numbers.
a(6) >= 720. - Michel Marcus, Nov 14 2021
From David A. Corneth, Nov 14 2021: (Start)
a(6) >= 1440. If a(6) > 1440 then a(6) > 50000.
a(7) >= 2400. If a(7) > 2400 then a(7) > 50000.
a(8) >= 7440. If a(8) > 7440 then a(8) > 100000.
a(9) >= 25920. If a(9) > 25920 then a(9) > 100000. (End)
Indeed, all these bounds are the corresponding values of a(6), a(7), a(8), a(9). Proof in link. For n >= 5, the five known terms are divisible by 240. - Bernard Schott, Nov 19 2021

			For n = 2, integers 1 and 2 are 2-phobe, then for m >= 3, every m = 1 + (m-1) with 1 < m-1 and 1 divides m-1, so, each m >= 3 is 2-phile number and a(2) = 2.

Diophante, A496 - Pentaphiles et pentaphobes (in French).
Bernard Schott, Proof that a(6) = 1440 and other proofs.

Cf. A349189.
k-phile numbers: A160811 \ {5} (k=3), A348517 (k=4), A348518 (k=5).
k-phobe numbers: A019532 (k=3), A348519 (k=4), A348520 (k=5).

a(6)-a(11) from David A. Corneth, Nov 19 2021

cp's OEIS Frontend

A349188 Largest n-phobe number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions