A342042 - OEIS

%I #94 Dec 01 2024 14:39:27

%S 0,1,2,3,4,5,6,7,8,9,10,11,12,30,13,14,50,15,16,70,17,18,90,19,23,24,

%T 51,25,26,71,27,28,91,29,31,32,33,34,52,35,36,72,37,38,92,39,45,46,73,

%U 47,48,93,49,53,54,55,56,74,57,58,94,59,67,68,95,69,75,76,77,78,96,79,89,97,98,99,101

%N When a digit d in the digit-stream of this sequence is even, the next digit is > d.

%C The definition refers to the digit-stream in the sequence (ignoring the commas), which starts 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 3, 0, ...

%C The sequence is always extended with the smallest nonnegative integer not yet present that doesn't lead to a contradiction.

%C Conjecture: This contains every nonnegative integer except those in A347298. - _N. J. A. Sloane_, Aug 26 2021

%C This is a theorem (proved in September 2021, see link for a new, more detailed proof). - _Sebastian Karlsson_, Nov 28 2024. See the other link for an alternative, shorter, proof. - _N. J. A. Sloane_, Nov 29 2024

%C Comment from _N. J. A. Sloane_, Dec 01 2024 (Start)

%C Let c1 = 7.422574840... and c2 = 1.3824387... be the constants defined in A377918. Then the k-th meeting point in the present sequence, where the branches come together, is close to (c2*c1^k, 10^k).

%C   Let x = c2*c1^k, and express k in terms of x.

%C Then this point has coordinates (x,y) where y = (x/c2)^c3, with c3 = (log 10)/(log c1) = 1.14869... This defines a curve that is a good approximation to the lower envelope of the present sequence.

%C For example, the fifth meeting point has coordinates (31148, 101010) (see A377918) and the formula here gives (x,y) = (31148, 100003.0039).

%C (End)

%H N. J. A. Sloane, <a href="/A342042/b342042.txt">Table of n, a(n) for n = 1..20000</a> [Computed using Rémy Sigrist's PARI program. The first 10000 terms were computed by Peter Kagey]

%H Sebastian Karlsson, <a href="/A342042/a342042.pdf">Proof of Theorem</a>.

%H Rémy Sigrist, <a href="/A342042/a342042.png">Colored scatterplot of the first 20000 terms</a> (where the color is function of the leading digit of a(n))

%H Rémy Sigrist, <a href="/A342042/a342042.gp.txt">PARI program for A342042</a>

%H N. J. A. Sloane, <a href="https://www.youtube.com/watch?v=3RAYoaKMckM">A Nasty Surprise in a Sequence and Other OEIS Stories</a>, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; <a href="https://sites.math.rutgers.edu/~zeilberg/expmath/sloane85BD.pdf">Slides</a> [Mentions this sequence]

%H N. J. A. Sloane, <a href="/A342042/a342042_2.txt">An Alternative Proof of the Theorem.</a>

%o (PARI) \\ See Links section.

%o (Python)

%o def cond(s, minfirst):

%o     return all(s[i+1] > s[i] for i in range(len(s)-1) if s[i] in "02468")

%o def aupton(terms):

%o     alst, seen = [0], {0}

%o     while len(alst) < terms:

%o         d = alst[-1]%10

%o         an = minfirst = (1 - d%2)*(d+1)

%o         stran = str(an)

%o         while an in seen or not cond(stran, minfirst):

%o             an += 1

%o             stran = str(an)

%o             if int(stran[0]) < minfirst:

%o                 an = minfirst*10**(len(stran)-1)

%o         alst.append(an); seen.add(an)

%o     return alst

%o print(aupton(77)) # _Michael S. Branicky_, Sep 07 2021

%Y Cf. A342043, A342044, A342045, A342046 and A342047 (variations on the same idea).

%Y See A377913 and A377914 for records.

%Y See also A347298.

%K base,nonn,nice,changed

%O 1,3

%A _Eric Angelini_, Feb 26 2021

%E Edited by _N. J. A. Sloane_, Nov 24 2024