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

%I A381137 #24 Mar 12 2025 04:14:49
%S A381137 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,
%T A381137 32,33,34,35,36,37,38,39,41,43,44,46,47,48,49,50,51,52,53,55,57,58,59,
%U A381137 60,61,62,63,64,65,67,68,69,70,71,73,74,76,79,81,82,83,85,86
%N A381137 Lexicographically earliest sequence of distinct positive integers such that no 3 terms are in harmonic progression.
%C A381137 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.
%C A381137 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.
%C A381137 Every prime occurs in the sequence.
%H A381137 Neal Gersh Tolunsky, <a href="/A381137/b381137.txt">Table of n, a(n) for n = 1..10000</a>
%e A381137 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.
%o A381137 (Python)
%o A381137 from itertools import count
%o A381137 def A381137_generator():
%o A381137     a_list = []
%o A381137     forbidden = set()
%o A381137     a = 0
%o A381137     while 1:
%o A381137         a = next(k for k in count(a+1) if k not in forbidden)
%o A381137         yield a
%o A381137         forbidden.update(a*b//m for b in a_list if (m:=2*b-a) > 0 and a*b%m == 0)
%o A381137         a_list.append(a) # _Pontus von Brömssen_, Mar 04 2025
%Y A381137 Analogous sequences: A003278 (for arithmetic progressions), A000452 (for geometric progressions).
%Y A381137 Cf. A005279, A174905.
%K A381137 nonn
%O A381137 1,2
%A A381137 _Neal Gersh Tolunsky_, Feb 15 2025