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Each of the ten sequences below have a similarly worded rule that determines the next number in the sequence. Try to figure the rules and subsequent numbers for each of the sequences. They are not necessarily ordered from simplest to most complex.

  1. 1, 2, 4, 8, 16, 23, 28, 38, 49, 62, ...
  2. 1, 2, 4, 7, 9, 12, 18, 24, 32, 38, ...
  3. 1, 4, 8, 13, 21, 30, 36, 45, 54, 63, ...
  4. 1, 16, 35, 55, 61, 80, 85, 90, 104, 119, ...
  5. 1, 7, 14, 17, 24, 32, 40, 43, 46, 49, ...
  6. 1, 10, 15, 19, 25, 30, 35, 40, 44, 48, ...
  7. 1, 2, 3, 5, 7, 10, 12, 14, 17, 19, ...
  8. 1, 3, 8, 15, 22, 32, 42, 51, 58, 70, ...
  9. 1, 2, 4, 7, 11, 14, 19, 24, 31, 38, ...
  10. 1, 2, 4, 6, 10, 12, 18, 25, 30, 36, ...

Hint (for number 10):

This sequence is the most different; neither math nor outside knowledge is useful here. Thankfully, you already have everything need to solve it.

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  • $\begingroup$ rot13(V fbyirq gur frpbaq, guveq, friragu, rvtugu naq avagu. V nyfb unir gur svefg ohg bayl jura gur ynfg ryrzrag vf 62. Qb lbh jnag zr gb fubj abj be jura V fbyir gur bguref?) $\endgroup$
    – Michał Wójcik
    Commented Jan 29, 2020 at 15:20
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    $\begingroup$ @MichałWójcik rot13(V'yy bayl npprcg n fbyhgvba sbe nyy bs gurz, ohg vs lbh jnag gb qbphzrag jung lbh'ir fbyirq srry serr gb. Vg'yy cebonoyl uryc bgure chmmyr fbyiref, gbb. Nyfb avpr pngpu ba gur svefg frdhrapr, gur ynfg ahzore unq n glcb. V'ir qbhoyr-purpxrq gur bgure frdhraprf sbe reebef nf jryy.) $\endgroup$
    – legodude5000
    Commented Jan 29, 2020 at 16:20
  • $\begingroup$ rot13(V jvyy svtug n yvggyr zber, ohg vs V tvir hc, gura V jvyy choyvfu na vapbzcyrgr yvfg. Svsgu, fvkgu naq gragu ner fgvyy yrsg. Vf tbvat orlbaq Ratyvfu arprffnel?) $\endgroup$
    – Michał Wójcik
    Commented Jan 29, 2020 at 18:00
  • $\begingroup$ rot13(Abcr, Ratyvfu vf gur bayl ynathntr vg jbhyq uryc gb xabj. Vs lbh jnag n zvabe uvag yrg zr xabj naq V pna fcbvyre bar.) $\endgroup$
    – legodude5000
    Commented Jan 30, 2020 at 8:31
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    $\begingroup$ @legodude5000 its been over 10 days and No.10 is still bugging me. As much as I would hate to admit it I think I need a hint $\endgroup$
    – Nobody
    Commented Feb 10, 2020 at 8:46

2 Answers 2

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I solved 8 of them by myself. I was inspired by the @Nobody's answer with the 5th and 6th one. I've already completed all the solutions:

1.

a(n+1) = a(n) + sum of digits of a(n)

2.

a(n+1) = a(n) + number of divisors of a(n)

3.

a(n+1) = a(n) + number of letters in english of a(n)

4.

a(n+1) = a(n) + position of the first letter in english alphabet of a(n)

5.

a(n+1) = a(n) + phonepad number (T9 encoding) of the first letter of a(n)

6.

a(n+1) = a(n) + number of column on QWERTY keyboard of the first letter of a(n)

7.

a(n+1) = a(n) + number of 1's in binary representation of a(n)

8.

a(n+1) = a(n) + number of highlighted segments in 7-segment display of a(n)

9.

a(n+1) = a(n) + number of straight lines in Roman numerals notation of a(n)

10.

a(n+1) = a(n) + lowest position of a(n) in any of ten sequences
2 = 1 + 1 $\leftarrow$ 1 is on the 1st position in all of sequences
4 = 2 + 2 $\leftarrow$ 2 is on the 2nd position in sequences No. 1, 2, 7, 9 and 10
6 = 4 + 2 $\leftarrow$ 4 is on the 2nd position in sequence No. 3
10 = 6 + 4 $\leftarrow$ 6 is on the 4th position in sequences No. 10
12 = 10 + 2 $\leftarrow$ 10 is on the 2nd position in sequences No. 6
18 = 12 + 6 $\leftarrow$ 12 is on the 6th position in sequences No. 2 and 10
25 = 18 + 7 $\leftarrow$ 18 is on the 7th position in sequences No. 2 and 10
30 = 25 + 5 $\leftarrow$ 25 is on the 5th position in sequences No. 6
36 = 30 + 6 $\leftarrow$ 30 is on the 6th position in sequences No. 3 and 6
So the next one will be 43.
43 = 36 + 7 $\leftarrow$ 36 is on the 7th position in sequences No. 3

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Just putting down ideas/solutions as I think of them (Have all)

  1. here the sequence is the previous term plus the sum of the digits in the previous term
    so the next term should be 62 + 8 = 70 [6 + 2 = 8]

  2. here the sequence is the previous term plus the number of its divisors in the previous term
    so the next term should be 38 + 4 = 42 [38 has 4 divisors - {1, 2, 19, 38}]

  3. here the sequence is the previous term plus the number of letters in its spelling
    so here the next term would be 63 + 10 = 73 [sixty(5) three(5)]

  4. here the sequence is the previous term plus the number of the position of the first letter in its spelling
    so here the next term would be 119 + 15 = 134 [One hundred and nineteen, O is the 15th alphabet]

  5. here the sequence is the previous term plus the number on which the first character of its spelling is on a phonepad
    so here the next term would be 49 + 3 = 52[forty nine - F is on 3 on a landline]

  6. here the sequence is the previous term plus the number of the column on which the first character of its spelling is on a qwerty keyboard
    so here the next term would be 48 + 4 = 52[forty eight - F is on the same column as 4 on a qwerty keyboard]

  7. here the sequence is the previous term plus the number ones in its binary representation
    so here the next term would be 19 + 3 = 22 [10011]

  8. here the sequence is the previous term plus the number segments in its calculator representation (also used in some elevators, I don't know what the term is called)
    so here the next term would be 70 + 9 = 79 [7(3) 0(6)]

  9. here the sequence is the previous term plus the number of lines in its roman representation
    so here the next term would be 38 + 11 = 49 [XXXVIII]

Got number 10 after the hint

  1. here the sequence is the previous term plus the lowest position of its occurrence in the given sequences
    so here the next term would be 36 + 7 = 43 [in sequence 3, 36 is the 7th number]

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  • $\begingroup$ rot13(Nyy pbeerpg fb sne! Gur grez lbh'er ybbxvat sbe va ahzore rvtug vf 'frira frtzrag qvfcynl'.) $\endgroup$
    – legodude5000
    Commented Jan 31, 2020 at 10:54
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    $\begingroup$ @legodude5000 ahh thank you. Still working on the others. Its a nice puzzle that has me thinking about the oddest things to figure out the sequences. I'm sure its going to be something obvious once someone points it out. $\endgroup$
    – Nobody
    Commented Jan 31, 2020 at 10:56
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    $\begingroup$ @legodude5000 added a guess for No.5 and No.6 still thinking of No.10. and yes it was obvious once i found the answers $\endgroup$
    – Nobody
    Commented Feb 2, 2020 at 16:45
  • $\begingroup$ Great work! rot13(Grpuavpnyyl ahzore 10'f ehyr vapyhqrf nyy gur frdhraprf; r.t. 6 arire nccrnef va gur svefg avar frdhraprf ohg vf sbhegu va frdhrapr gra.) I'm not sure whether to accept you or @Michał Wójcik; You had the first answer with the most correct sequences at the time, but Michał finished the entire puzzle first. Not sure what the SO etiquette is for situations like this $\endgroup$
    – legodude5000
    Commented Feb 11, 2020 at 7:49
  • $\begingroup$ @legodude5000 im fine with Michal getting the check, he had solved the 7 patterns before me (according to his comments) and did solve the last one before me as well $\endgroup$
    – Nobody
    Commented Feb 11, 2020 at 7:52

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