A381137 Lexicographically earliest sequence of distinct positive integers such that no 3 terms are in harmonic progression.

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 73, 74, 76, 79, 81, 82, 83, 85, 86

Offset: 1

Author

Neal Gersh Tolunsky, Feb 15 2025

Keywords

• nonn

Comments

A harmonic progression is a sequence of values whose reciprocals are in arithmetic progression. Equivalently, if (a, b, c) is a harmonic progression, then b is the harmonic mean of a and c.

a(n) is the smallest integer greater than a(n-1) which does not form a 3-term harmonic progression with 2 previously occurring terms.

Every prime occurs in the sequence.

Examples

			6 is not a term in the sequence because it would form a harmonic progression with 2 and 3, which occurred earlier. The progression (1/6, 1/3, 1/2) has common difference 1/6.
		

Links

• Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000

Crossrefs

Analogous sequences: A003278 (for arithmetic progressions), A000452 (for geometric progressions).

Cf. A005279, A174905.

Programs

Python

from itertools import count
def A381137_generator():
    a_list = []
    forbidden = set()
    a = 0
    while 1:
        a = next(k for k in count(a+1) if k not in forbidden)
        yield a
        forbidden.update(a*b//m for b in a_list if (m:=2*b-a) > 0 and a*b%m == 0)
        a_list.append(a) # Pontus von Brömssen, Mar 04 2025