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A377351 Lexicographically earliest sequence of positive integers such that the means of consecutive terms are all distinct.

%I A377351 #11 Oct 28 2024 16:24:40
%S A377351 1,2,4,7,5,10,12,18,13,16,28,23,14,25,48,32,17,46,30,45,67,60,27,71,
%T A377351 39,85,68,99,78,44,102,87,126,57,118,69,121,74,125,119,112,136,107,
%U A377351 110,170,120,175,142,194,75,222,152,164,180,177,184,188,135,255,210
%N A377351 Lexicographically earliest sequence of positive integers such that the means of consecutive terms are all distinct.
%C A377351 In other words, for any distinct nonempty intervals t..u and v..w, Sum_{i = t..u} a(i)/(u-t+1) <> Sum_{j = v..w} a(j)/(w-v+1).
%C A377351 This sequence corresponds essentially to the first differences of A033808.
%C A377351 By necessity, all terms are distinct.
%H A377351 Rémy Sigrist, <a href="/A377351/b377351.txt">Table of n, a(n) for n = 1..10000</a>
%H A377351 Rémy Sigrist, <a href="/A377351/a377351.txt">C++ program</a>
%F A377351 a(n) = A033808(n) - A033808(n-1).
%e A377351 The first terms, alongside the means of consecutive terms ending with a(n), are:
%e A377351   n  a(n)  Corresponding means
%e A377351   -  ----  ------------------------------------------
%e A377351   1     1  1
%e A377351   2     2  3/2, 2
%e A377351   3     4  7/3, 3, 4
%e A377351   4     7  7/2, 13/3, 11/2, 7
%e A377351   5     5  19/5, 9/2, 16/3, 6, 5
%e A377351   6    10  29/6, 28/5, 13/2, 22/3, 15/2, 10
%e A377351   7    12  41/7, 20/3, 38/5, 17/2, 9, 11, 12
%e A377351   8    18  59/8, 58/7, 28/3, 52/5, 45/4, 40/3, 15, 18
%o A377351 (C++) // See Links section.
%o A377351 (Python)
%o A377351 from fractions import Fraction
%o A377351 from itertools import count, islice
%o A377351 def agen(): # generator of terms
%o A377351     alst, means_seen = [1], {1}
%o A377351     while True:
%o A377351         yield alst[-1]
%o A377351         for k in count(1):
%o A377351             if k in means_seen: continue
%o A377351             mk, failed, sk = {k}, False, k
%o A377351             for j in range(1, len(alst)+1):
%o A377351                 sk += alst[-j]
%o A377351                 m = Fraction(sk, j+1)
%o A377351                 if m in means_seen or m in mk: failed = True; break
%o A377351                 mk.add(m)
%o A377351             if not failed: break
%o A377351         means_seen |= mk
%o A377351         alst.append(k)
%o A377351 print(list(islice(agen(), 60))) # _Michael S. Branicky_, Oct 26 2024, Oct 28 2024
%Y A377351 Cf. A033808, A377388.
%K A377351 nonn
%O A377351 1,2
%A A377351 _Rémy Sigrist_, Oct 26 2024