Problems & Puzzles: Puzzles

Puzzle 980. The "Commas" sequence.

Here we make some simple questions related to the OEIS sequence A121805, original by Belgian journalist Eric Angelini. See it here too.

As perhaps you already know this sequence is composed by exactly 2137453 members.

If we compute and count the absolute difference D (0<=D<100) between consecutive members of A121805, we may expect that the count will be zero for D@10=0. But unexpectedly the count is zero also for the following two sets of D values:

Set 1, D={15, 65}
Set 2, D={4, 24, 44, 64 & 84}

(See below the Counting Table, in descending order)

Q1. Any idea why this happens?

In the opposite counting side, the maximal count (166669) occurs in the values D=5 & 55.

Q2. Any idea why this happens?

On other issue, A121805 arbitrarily starts in the integer "1". What happens if the first term is "2" instead "1". This new sequence is described in A139284.

a) It appears that this sequence and A121805 have no terms in common

b) Furthermore, this sequence exists for at least 1551000000 terms.

The second issue means that no final term has been found, as per Jul 22 2008.

Q3. Can you find the final term to A139284 (or at least extend the limit of terms computed)?

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Counting Table for Q1 & Q2

 D Count 5 166669 55 166669 2 47624 22 47624 42 47624 62 47624 82 47624 88 46255 8 46254 28 46254 48 46254 68 46254 14 41152 74 41152 94 41152 34 41151 54 41151 56 39291 16 39290 96 39290 36 39289 76 39289 71 24156 11 24155 41 24155 51 24155 81 24155 1 24154 21 24154 31 24154 91 24154 61 24153 13 23149 23 23149 43 23149 63 23149 83 23149 3 23148 53 23148 73 23148 93 23148 33 23147 77 21369 7 21368 17 21368 37 21368 47 21368 67 21368 27 21367 57 21367 97 21367 87 21366 89 20578 69 20577 99 20577 9 20576 19 20576 29 20576 39 20576 49 20575 59 20575 79 20575 45 7143 95 7143 52 4270 12 4269 32 4269 72 4269 92 4269 35 918 85 916 6 479 46 479 86 479 26 477 66 477 18 35 58 35 78 35 38 34 98 34 25 10 75 10 4 0 10 0 15 0 20 0 24 0 30 0 40 0 44 0 50 0 60 0 64 0 65 0 70 0 80 0 84 0 90 0

All contributions from the week 23-30 November 2019, are from: Giovanni Resta, Oscar Volpatti, Simon Cavegn.

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Giovanni wrote:

Concerning question Q3, I think that the last term of sequence A139284 is:
a(194697747222394) = 9999999999999918. (24/11/2019)

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Oscar wrote:

The OEIS sequence A139284 is composed by exactly 194697747222394 members:
a(194697747222394) = 9999999999999918,
a(194697747222395) doesn't exist. (29/11/2019)

[This is a second and independent computation of the quantity of terms of A139284, CR ]

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Simon wrote:

Q3 Did not find the end yet. Here some data to double check my progress:

n=10 a=319

n=100 a=4781
n=1000 a=48086
n=10000 a=460902
n=100000 a=4722879
n=1000000 a=46056640
n=10000000 a=458881540
n=100000000 a=4982244123
n=1000000000 a=49280750965
n=10000000000 a=502362532122
n=100000000000 a=4632741211120
n=1000000000000 a=47065625430561
n=10000000000000 a=472525419983779
n=18442867171548 a=882813797309036

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