What comes next in these sequences?

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In summary, the first sequence follows the pattern of 2 times the nth prime number minus the (n+1)th prime number. The second sequence is the smallest number with n distinct factors that is larger than the previous term. Both sequences involve thinking about prime numbers and factors.
  • #1
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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  • #2
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
 
  • #3
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
 
  • #4
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
 
  • #5
Is the first sequence:

a(n) = 2* nth prime number - (n+1)th prime number ?
 
  • #6
siddharth said:
Is the first sequence:

a(n) = 2* nth prime number - (n+1)th prime number ?

Bingo!

DaveE
 

1. What are two more sequences?

Two more sequences refer to two additional sets of numbers or items that follow a specific pattern or sequence.

2. How are two more sequences related to each other?

Two more sequences can be related in various ways, such as having the same starting point or following a similar pattern.

3. What is the purpose of studying two more sequences?

The purpose of studying two more sequences is to identify patterns, make predictions, and potentially solve problems or equations related to the sequences.

4. How can two more sequences be represented?

Two more sequences can be represented in various ways, such as using a table, graph, or algebraic expression.

5. Are there any real-life examples of two more sequences?

Yes, there are many real-life examples of two more sequences, such as predicting stock market trends, calculating population growth, and analyzing weather patterns.

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