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3 I stuttered!
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The requested numbers are

$$\mathbf{158,~~ 190},~~ 212,~~ 224,~~ 246,~~ 278,~~ 310,~~ 322,~~ \mathbf{354,~~ 376}$$

The formula for the n th Uncle number, $U_n$, is

$$U_n = 5\times p_n + 2\times n - 19$$ where $p_n$ is the n th prime.  The numbers shown in the question are $U_{13} = 5\times41 + 2\times13 - 19 = 212$  through  $U_{18} = 5\times61 + 2\times18 - 19 = 322$

For completeness, I could explain that $U_{1} ~~= 5\times2~ ~+ 2\times1~ ~- 19 = 5\times2 ~+~ 2\times1 ~-~ 19 = 10 + 2 - 19 = -7$
$U_{1} ~~= 5\times2~ ~+ 2\times1~ ~- 19 = 10 + 2 - 19 = -7$

How did I figure that out?

  • We have a monotonically increasing sequence of integers that begins with an odd number, but it appears to settle into a groove of being all even numbers.  That made me think: that’s one bit off from a monotonically increasing sequence of integers that begins with an even number, but settles into a groove of being all odd numbers.

  • Hint #3 mentions some parameters regarding grandmother’s home.

And, having solved the problem and written the solution, I took another look and noticed that

  • the question body mentions the word “prime”:
    … watching a prime time show.
    Looking at the question’s edit history, I see that Hint #1 originally said, “Is in the first para.  Tricky and you need to look around.” but that was edited out.

The requested numbers are

$$\mathbf{158,~~ 190},~~ 212,~~ 224,~~ 246,~~ 278,~~ 310,~~ 322,~~ \mathbf{354,~~ 376}$$

The formula for the n th Uncle number, $U_n$, is

$$U_n = 5\times p_n + 2\times n - 19$$ where $p_n$ is the n th prime.  The numbers shown in the question are $U_{13} = 5\times41 + 2\times13 - 19 = 212$  through  $U_{18} = 5\times61 + 2\times18 - 19 = 322$

For completeness, I could explain that $U_{1} ~~= 5\times2~ ~+ 2\times1~ ~- 19 = 5\times2 ~+~ 2\times1 ~-~ 19 = 10 + 2 - 19 = -7$

How did I figure that out?

  • We have a monotonically increasing sequence of integers that begins with an odd number, but it appears to settle into a groove of being all even numbers.  That made me think: that’s one bit off from a monotonically increasing sequence of integers that begins with an even number, but settles into a groove of being all odd numbers.

  • Hint #3 mentions some parameters regarding grandmother’s home.

And, having solved the problem and written the solution, I took another look and noticed that

  • the question body mentions the word “prime”:
    … watching a prime time show.
    Looking at the question’s edit history, I see that Hint #1 originally said, “Is in the first para.  Tricky and you need to look around.” but that was edited out.

The requested numbers are

$$\mathbf{158,~~ 190},~~ 212,~~ 224,~~ 246,~~ 278,~~ 310,~~ 322,~~ \mathbf{354,~~ 376}$$

The formula for the n th Uncle number, $U_n$, is

$$U_n = 5\times p_n + 2\times n - 19$$ where $p_n$ is the n th prime.  The numbers shown in the question are $U_{13} = 5\times41 + 2\times13 - 19 = 212$  through  $U_{18} = 5\times61 + 2\times18 - 19 = 322$

For completeness, I could explain that
$U_{1} ~~= 5\times2~ ~+ 2\times1~ ~- 19 = 10 + 2 - 19 = -7$

How did I figure that out?

  • We have a monotonically increasing sequence of integers that begins with an odd number, but it appears to settle into a groove of being all even numbers.  That made me think: that’s one bit off from a monotonically increasing sequence of integers that begins with an even number, but settles into a groove of being all odd numbers.

  • Hint #3 mentions some parameters regarding grandmother’s home.

And, having solved the problem and written the solution, I took another look and noticed that

  • the question body mentions the word “prime”:
    … watching a prime time show.
    Looking at the question’s edit history, I see that Hint #1 originally said, “Is in the first para.  Tricky and you need to look around.” but that was edited out.

2 Demonstrated that my series begins with −7.
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The requested numbers are

$$\mathbf{158,~~ 190},~~ 212,~~ 224,~~ 246,~~ 278,~~ 310,~~ 322,~~ \mathbf{354,~~ 376}$$

The formula for the n th Uncle number, $U_n$, is

$$U_n = 5\times p_n + 2\times n - 19$$ where $p_n$ is the n th prime.  The numbers shown in the question are $U_{13} = 5\times41 + 2\times13 - 19 = 212$  through  $U_{18} = 5\times61 + 2\times18 - 19 = 322$

For completeness, I could explain that $U_{1} ~~= 5\times2~ ~+ 2\times1~ ~- 19 = 5\times2 ~+~ 2\times1 ~-~ 19 = 10 + 2 - 19 = -7$

How did I figure that out?

  • We have a monotonically increasing sequence of integers that begins with an odd number, but it appears to settle into a groove of being all even numbers.  That made me think: that’s one bit off from a monotonically increasing sequence of integers that begins with an even number, but settles into a groove of being all odd numbers.

  • Hint #3 mentions some parameters regarding grandmother’s home.

And, having solved the problem and written the solution, I took another look and noticed that

  • the question body mentions the word “prime”:
    … watching a prime time show.
    Looking at the question’s edit history, I see that Hint #1 originally said, “Is in the first para.  Tricky and you need to look around.” but that was edited out.

The requested numbers are

$$\mathbf{158,~~ 190},~~ 212,~~ 224,~~ 246,~~ 278,~~ 310,~~ 322,~~ \mathbf{354,~~ 376}$$

The formula for the n th Uncle number, $U_n$, is

$$U_n = 5\times p_n + 2\times n - 19$$ where $p_n$ is the n th prime.  The numbers shown in the question are $U_{13} = 5\times41 + 2\times13 - 19 = 212$  through  $U_{18} = 5\times61 + 2\times18 - 19 = 322$

How did I figure that out?

  • We have a monotonically increasing sequence of integers that begins with an odd number, but it appears to settle into a groove of being all even numbers.  That made me think: that’s one bit off from a monotonically increasing sequence of integers that begins with an even number, but settles into a groove of being all odd numbers.

  • Hint #3 mentions some parameters regarding grandmother’s home.

And, having solved the problem and written the solution, I took another look and noticed that

  • the question body mentions the word “prime”:
    … watching a prime time show.
    Looking at the question’s edit history, I see that Hint #1 originally said, “Is in the first para.  Tricky and you need to look around.” but that was edited out.

The requested numbers are

$$\mathbf{158,~~ 190},~~ 212,~~ 224,~~ 246,~~ 278,~~ 310,~~ 322,~~ \mathbf{354,~~ 376}$$

The formula for the n th Uncle number, $U_n$, is

$$U_n = 5\times p_n + 2\times n - 19$$ where $p_n$ is the n th prime.  The numbers shown in the question are $U_{13} = 5\times41 + 2\times13 - 19 = 212$  through  $U_{18} = 5\times61 + 2\times18 - 19 = 322$

For completeness, I could explain that $U_{1} ~~= 5\times2~ ~+ 2\times1~ ~- 19 = 5\times2 ~+~ 2\times1 ~-~ 19 = 10 + 2 - 19 = -7$

How did I figure that out?

  • We have a monotonically increasing sequence of integers that begins with an odd number, but it appears to settle into a groove of being all even numbers.  That made me think: that’s one bit off from a monotonically increasing sequence of integers that begins with an even number, but settles into a groove of being all odd numbers.

  • Hint #3 mentions some parameters regarding grandmother’s home.

And, having solved the problem and written the solution, I took another look and noticed that

  • the question body mentions the word “prime”:
    … watching a prime time show.
    Looking at the question’s edit history, I see that Hint #1 originally said, “Is in the first para.  Tricky and you need to look around.” but that was edited out.

1
source | link

The requested numbers are

$$\mathbf{158,~~ 190},~~ 212,~~ 224,~~ 246,~~ 278,~~ 310,~~ 322,~~ \mathbf{354,~~ 376}$$

The formula for the n th Uncle number, $U_n$, is

$$U_n = 5\times p_n + 2\times n - 19$$ where $p_n$ is the n th prime.  The numbers shown in the question are $U_{13} = 5\times41 + 2\times13 - 19 = 212$  through  $U_{18} = 5\times61 + 2\times18 - 19 = 322$

How did I figure that out?

  • We have a monotonically increasing sequence of integers that begins with an odd number, but it appears to settle into a groove of being all even numbers.  That made me think: that’s one bit off from a monotonically increasing sequence of integers that begins with an even number, but settles into a groove of being all odd numbers.

  • Hint #3 mentions some parameters regarding grandmother’s home.

And, having solved the problem and written the solution, I took another look and noticed that

  • the question body mentions the word “prime”:
    … watching a prime time show.
    Looking at the question’s edit history, I see that Hint #1 originally said, “Is in the first para.  Tricky and you need to look around.” but that was edited out.