What comes next in these sequences?

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Discussion Overview

The discussion revolves around identifying patterns and predicting the next numbers in two mathematical sequences. The focus includes exploratory reasoning and mathematical reasoning related to sequences and their properties.

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant presents two sequences and notes a surprising pattern followed by erratic behavior in the first sequence.
  • Another participant suggests that the first sequence may relate to prime numbers, while the second sequence might involve factors.
  • A participant proposes a formula for the second sequence, suggesting it represents the smallest number with n distinct factors, noting that odd-numbered terms are squares.
  • Another participant agrees with the proposed formula for the second sequence and refines it to specify that it should be the smallest number with n distinct factors larger than the previous term.
  • A participant proposes a formula for the first sequence involving prime numbers, which is later confirmed by another participant.

Areas of Agreement / Disagreement

Participants generally agree on the proposed formulas for both sequences, but the discussion remains open regarding the next numbers in the sequences, as no definitive conclusions are drawn about them.

Contextual Notes

The discussion does not resolve the next numbers in the sequences, and assumptions about the nature of the sequences are not fully explored.

Who May Find This Useful

Individuals interested in mathematical sequences, number theory, and pattern recognition may find this discussion relevant.

davee123
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Just futzing around, this sequence was suprisingly patterned for the first 8 numbers, then became erratic:

1, 1, 3, 3, 9, 9, 15, 15, 17, 27, ?

And for the sake of more fooling around, this one just popped into my head:

1, 2, 4, 6, 16, 18, 64, 66, 100, 112, ?

DaveE
 
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Hm... perhaps hints are in order?

davee123 said:
1, 1, 3, 3, 9, 9, 15, 15, 17, 27, ?

Hint: Think primes!
davee123 said:
1, 2, 4, 6, 16, 18, 64, 66, 100, 112, ?

Hint: Think factors!
(and you might not want to bother trying to find the next number in the sequence, if going by hand)

DaveE
 
is the second sequence something like:

a(n) = the smallest number with n distinct factors (this would explain the odd-numbered terms being squares)

havent looked at the first one yet, but the second jumped out at me
 
rhj23 said:
is the second sequence something like:

a(n) = the smallest number with n distinct factors (this would explain the odd-numbered terms being squares)

havent looked at the first one yet, but the second jumped out at me

Yep! Or, close enough. The smallest number with n distinct factors that's larger than a(n-1). Nice job! Now for the 1st one!

DaveE
 
Is the first sequence:

a(n) = 2* nth prime number - (n+1)th prime number [/color]?
 
siddharth said:
Is the first sequence:

a(n) = 2* nth prime number - (n+1)th prime number [/color]?

Bingo!

DaveE
 

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