A349189 Number of n-phobe numbers.

2, 9, 23, 68, 177, 459, 1162, 2947, 7306, 18202

Offset: 2

Author

Bernard Schott, Nov 14 2021

Keywords

• nonn
• more

Comments

A n-phile integer m is such that there are n positive integers b_1 < b_2 < ... < b_j < ... < b_n with the property 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 words 'n-phile' and 'n-phobe' come from the French website Diophante (see link).

The number of n-phobe numbers is always finite, the smallest one is always 1 and the largest n-phobe number is in A349188.

a(6) >= 176. - Michel Marcus, Nov 15 2021

a(6) >= 177, a(7) >= 459, a(8) >= 1162, a(9) >= 2947. - David A. Corneth, Nov 15 2021

Indeed, all these bounds are the corresponding values of a(6), a(7), a(8) and a(9). Proof comes from Proof link in A349188. - Bernard Schott, Nov 19 2021

Examples

			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.
		

Links

• Diophante, A496 - Pentaphiles et pentaphobes (in French).

Crossrefs

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).

Cf. A349188.

Extensions

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