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A360599 Ratios of consecutive terms of A360598: a(n) = max(A360598(n), A360598(n+1)) / min(A360598(n), A360598(n+1)).

%I A360599 #11 Feb 14 2023 12:55:02
%S A360599 1,2,3,6,4,5,20,7,8,56,9,10,90,11,12,132,13,14,182,15,16,240,17,18,
%T A360599 306,19,21,399,22,23,506,24,25,600,26,27,702,28,29,812,30,31,930,32,
%U A360599 33,1056,34,35,1190,36,37,1332,38,39,1482,40,41,1640,42,43,1806,44
%N A360599 Ratios of consecutive terms of A360598: a(n) = max(A360598(n), A360598(n+1)) / min(A360598(n), A360598(n+1)).
%C A360599 This sequence is a permutation of the positive integers with inverse A360600:
%C A360599 - we already know that all terms are distinct, so we just have to show that all integers appear,
%C A360599 - by contradiction: let r be the least value missing from this sequence,
%C A360599 - once the values 1..r-1 have appeared in this sequence, the sequence A360598 can only decrease finitely many times,
%C A360599 - the next increase in A360598 will correspond to the ratio r.
%H A360599 Rémy Sigrist, <a href="/A360599/a360599.gp.txt">PARI program</a>
%H A360599 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A360599 For n = 15:
%e A360599     A360598(15) = 11 and A360598(16) = 132,
%e A360599     so a(15) = 132 / 11 = 12.
%e A360599 For n = 16:
%e A360599     A360598(16) = 132 and A360598(17) = 1,
%e A360599     so a(16) = 132 / 1 = 132.
%o A360599 (PARI) See Links section.
%o A360599 (Python)
%o A360599 from itertools import islice
%o A360599 def agen(): # generator of terms
%o A360599     an, ratios = 1, set()
%o A360599     while True:
%o A360599         k = 1
%o A360599         q, r = divmod(max(k, an), min(k, an))
%o A360599         while r != 0 or q in ratios:
%o A360599             k += 1
%o A360599             q, r = divmod(max(k, an), min(k, an))
%o A360599         an = k
%o A360599         ratios.add(q)
%o A360599         yield q
%o A360599 print(list(islice(agen(), 66))) # _Michael S. Branicky_, Feb 13 2023
%Y A360599 Cf. A360597, A360598, A360600 (inverse).
%K A360599 nonn
%O A360599 1,2
%A A360599 _Rémy Sigrist_, Feb 13 2023