[seqfan] Re: The Annoyance Sequence

Thu Jul 20 15:38:25 CEST 2023

Could it be that 5 is the only number that never appears?

On Thu, Jul 20, 2023, 9:21 AM Ali Sada <pemd70 at yahoo.com> wrote:

> Hi Allan,
>
> These are the terms I got for this version
>
> 3, 2, 6, 4, 9, 10, 12, 13, 14, 15, 17, 7, 19, 8, 20, 22, 24, 25, 27, 11,
> 16, 30, 32, 18, 34, 36, 37, 21, 40, 41, 42, 44, 45, 26, 48, 49, 50, 52, 29,
> 54, 31, 56, 58, 23, 33, 61, 35, 63, 65, 67, 38, 69, 28, 70, 72, 74, 75, 43,
> 78, 79, 81, 82, 83, 84, 86, 87, 89, 51, 91, 93, 53, 95, 97, 98, 99, 39, 57,
> 102, 104, 105, 106, 60, 109, 110, 111, 113, 114, 115, 46, 117, 118, 119,
> 120, 68, 123, 124, 126, 71, 128, 130, 131, 132, 134, 76, 136, 77, 138, 139,
> 141, 80, 143, 145, 147, 148, 59, 151, 85, 153, 155, 47, 88, 62, 158, 90,
> 161, 162, 163, 164, 165, 94, 167, 168, 170, 171, 172, 174, 176, 177, 178,
> 179, 101, 181, 183, 103, 73, 186, 187, 189, 107, 191, 108, 194, 195, 197,
> 199, 112, 201, 203, 204, 206, 207, 208, 210, 212, 64, 214, 216, 121, 218,
> 66, 219, 221, 222, 223, 224, 226, 127, 228, 230, 129, 232, 234, 235, 92,
> 133, 238, 240, 241, 242, 244, 245, 96, 247, 249, 250, 251, 253, 142
>
> Best,
>
> Ali
>
> On Wednesday, July 19, 2023 at 10:10:21 PM GMT+1, Allan Wechsler <
> acwacw at gmail.com> wrote:
>
>
> I always try to make novel definitions simpler, and I came up with this
> variant of Ali Sada's idea.
>
> Start with 2,3,4,5,...
>
> At each step, "annoy" the first element, then record the new first element
> and throw it away. This gives:
>
> 3,4,2,5,6,... (pop 3)
> 2,5,6,7,4,8,9,... (pop 2)
> 6,7,4,8,9,5,10,11,... (pop 6)
>
> The popped numbers in order are 3,2,6,4,9,10,12,...
>
> I think this is not a permutation, because there is a collection of
> numbers that never manage to get emitted, of which I think the smallest is
> 5. This was all worked out by hand and I would appreciate confirmation.
>
> On Tue, Jul 18, 2023 at 1:30 AM Ali Sada via SeqFan <seqfan at list.seqfan.eu>
> wrote:
>
>  Thank you, Allan and Arthur, for answering my question, and for verifying
> the terms of the sequence. I really appreciate it. Under such temperature
> (40+ C) all the words came to my mind were negative (annoy, disturb,
> displace, etc.)
> And thank you, Mike, for the fascinating analysis. I wish you could have
> done it my version too! I would love to see the "cool" and the "late" ones!
>
> This is the definition I came up with:  Starting with a list of positive
> integers and setting a(1) = 1, we follow a process: remove each of the a(n)
> terms to the right of a(n), m, from the list, and then insert it m steps to
> the right. The number remaining to the right of a(n) becomes a(n+1).
>  I am sure it needs a lot of corrections and I would really appreciate the
> help!
> Best,
> Ali
>
>     On Sunday, July 16, 2023 at 07:10:06 PM GMT+1, Arthur O'Dwyer <
> arthur.j.odwyer at gmail.com> wrote:
>
>  Ali's description makes sense to me. (Although FWIW I don't like the
> "Annoying Sequence" name; it doesn't seem to convey anything useful about
> the structure, and it's just... annoying?)
> Here's a Python program that implements the idea (up to the 50th term):
>
>
>
> def smartlyTruncatedList(x):
>
>
>   i = len(x)
>
>   while (x[i-1] == i):
>
>     i -= 1
>
>   return x[:i]
>
>
>
>
> theList = list(range(1,10000))
>
> for i in range(0, 50):
>
>   print('Before step %d, the list is: %r' % (i+1,
> smartlyTruncatedList(theList)))
>
>   annoyingNumber = theList[i]
>
>   for d in range(0, annoyingNumber):
>
>     annoyedNumber = theList[i+1]
>
>
>
>     assert (i+2+annoyedNumber < len(theList))
>
>     theList = theList[:i+1] + theList[i+2:i+2+annoyedNumber] +
> [annoyedNumber] + theList[i+2+annoyedNumber:]
>
>
>
>     print('  After %d annoys %d, the list is: %r' % (annoyingNumber,
> annoyedNumber, smartlyTruncatedList(theList)))
>
>
>
> And here's the program's output, which agrees with Ali's:
>
>
>
> Before step 1, the list is: []
>
>   After 1 annoys 2, the list is: [1, 3, 4, 2]
>
> Before step 2, the list is: [1, 3, 4, 2]
>
>   After 3 annoys 4, the list is: [1, 3, 2, 5, 6, 7, 4]
>
>   After 3 annoys 2, the list is: [1, 3, 5, 6, 2, 7, 4]
>
>   After 3 annoys 5, the list is: [1, 3, 6, 2, 7, 4, 8, 5]
>
> Before step 3, the list is: [1, 3, 6, 2, 7, 4, 8, 5]
>
>   After 6 annoys 2, the list is: [1, 3, 6, 7, 4, 2, 8, 5]
>
>   After 6 annoys 7, the list is: [1, 3, 6, 4, 2, 8, 5, 9, 10, 11, 7]
>
>   After 6 annoys 4, the list is: [1, 3, 6, 2, 8, 5, 9, 4, 10, 11, 7]
>
>   After 6 annoys 2, the list is: [1, 3, 6, 8, 5, 2, 9, 4, 10, 11, 7]
>
>   After 6 annoys 8, the list is: [1, 3, 6, 5, 2, 9, 4, 10, 11, 7, 12, 8]
>
>   After 6 annoys 5, the list is: [1, 3, 6, 2, 9, 4, 10, 11, 5, 7, 12, 8]
>
> Before step 4, the list is: [1, 3, 6, 2, 9, 4, 10, 11, 5, 7, 12, 8]
>
>   After 2 annoys 9, the list is: [1, 3, 6, 2, 4, 10, 11, 5, 7, 12, 8, 13,
> 14, 9]
>
>   After 2 annoys 4, the list is: [1, 3, 6, 2, 10, 11, 5, 7, 4, 12, 8, 13,
> 14, 9]
>
> Before step 5, the list is: [1, 3, 6, 2, 10, 11, 5, 7, 4, 12, 8, 13, 14, 9]
>
> [...]
>
>
>
>   After 116 annoys 52, the list is: [1, 3, 6, 2, 10, 9, 5, 8, 22, 4, 13,
> 15, 20, 29, 7, 12, 18, 32, 59, 50, 19, 31, 14, 81, 16, 90, 17, 25, 78, 83,
> 21, 46, 65, 23, 41, 71, 64, 36, 53, 47, 58, 44, 35, 76, 62, 43, 88, 49,
> 123, 116, [...]
>
>
>
>
>
>
> On Sun, Jul 16, 2023 at 1:38 PM Ali Sada via SeqFan <seqfan at list.seqfan.eu>
> wrote:
>
>  Hi Alex and Yifan Xe,
> Thank you very much for your responses.
> Alex wrote  "I wonder if it's a permutation of the integers"
> I have seen similar comments on the OEIS regarding permutations of
> positive integers. TBH, I don't know why we need a proof in these cases.
> For example, in this sequence we started we the list of positive integers
> and the algorithm doesn't add or delete any numbers, so, the sequence
> should be a permutation of positive integers.
>
> The question is whether any specific integer is unlucky enough to keep
> getting bumped out of its position over and over and over forever.For
> example, we see that the number "77" doesn't appear anywhere in the first
> 50 terms of the sequence quoted above. Suppose "77" doesn't appear in the
> first 200 terms (as indeed it doesn't). Suppose it doesn't appear in the
> first billion terms. Suppose it never appears. If that's the case — if it's
> the case that "77" never appears in the OEIS sequence — then the OEIS
> sequence certainly can't be a permutation of the integers.
> HTH,Arthur
>
>
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