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A327743 a(n) = smallest positive number not already in the sequence such that for each k = 1, ..., n-1, the k-th differences are distinct.

%I A327743 #49 Oct 26 2019 04:41:34
%S A327743 1,2,4,3,6,11,5,9,7,13,10,18,8,15,27,14,23,12,22,17,28,16,29,20,34,19,
%T A327743 35,21,36,32,24,42,26,43,25,44,66,33,53,30,51,31,54,37,61,39,64,38,67,
%U A327743 40,70,41,68,47,75,50,76,45,77,49,80,48,81,46,82,52,86
%N A327743 a(n) = smallest positive number not already in the sequence such that for each k = 1, ..., n-1, the k-th differences are distinct.
%C A327743 Is this sequence a permutation of the positive integers?
%C A327743 Does each k-th difference contain all nonzero integers?
%C A327743 It is not difficult to show that if a(1), ..., a(k) satisfy the requirements, then any sufficiently large number is a candidate for a(k+1). So a(k) exists for all k. - _N. J. A. Sloane_, Sep 24 2019
%C A327743 The original definition was "Lexicographically earliest infinite sequence of distinct positive integers such that for every k >= 1, the k-th differences are distinct."
%C A327743 If only first differences are considered, one gets the classical Mian-Chowla sequence A005282. - _M. F. Hasler_, Oct 09 2019
%H A327743 Peter Kagey, <a href="/A327743/b327743.txt">Table of n, a(n) for n = 1..5000</a>
%e A327743 Illustration of the first eight terms of the sequence.
%e A327743 k | k-th differences
%e A327743 --+---------------------------------
%e A327743 0 |   1,  2,   4,   3,   6, 11, 5, 9
%e A327743 1 |   1,  2,  -1,   3,   5, -6, 4
%e A327743 2 |   1, -3,   4,   2, -11, 10
%e A327743 3 |  -4,  7,  -2, -13,  21
%e A327743 4 |  11, -9, -11,  34
%e A327743 5 | -20, -2,  45
%e A327743 6 |  18, 47
%e A327743 7 |  29
%t A327743 a[1] = 1;
%t A327743 a[n_] := a[n] = For[aa = Array[a, n-1]; an = 1, True, an++, If[FreeQ[aa, an], aa = Append[aa, an]; If[AllTrue[Range[n-1], Unequal @@ Differences[ aa, #]&], Return[an]]]];
%t A327743 a /@ Range[1, 100] (* _Jean-François Alcover_, Oct 26 2019 *)
%Y A327743 Cf. A175498.
%Y A327743 First differences: A327452; leading column of difference triangle: A327457.
%Y A327743 If ALL terms of the difference triangle must be distinct, see A327460 and A327762.
%Y A327743 Cf. A005282.
%K A327743 nonn,nice
%O A327743 1,2
%A A327743 _Peter Kagey_, Sep 24 2019
%E A327743 "Infinite" added to definition (for otherwise the one-term sequence 1 is earlier). - _N. J. A. Sloane_, Sep 25 2019
%E A327743 Changed definition to avoid use of "Lexicographically earliest infinite sequence" and the associated existence questions. - _N. J. A. Sloane_, Sep 28 2019