Sequencer | StrangerCoug's turn

[DeathRowKitty]

Can you clarify what you mean by that? The interpretation I'm taking out of it would exclude 55.

[popsofctown]

Yeah I messed up 55 doesn't work

EDIT:
No I just am easily suggestible.

11 is the largest factor of 55 less than 55.
5 is a factor of 55 beneficial to us.
11 - 5 = 6.
6 is a perfect number.

[popsofctown]

.

[DeathRowKitty]

Ohhh okay I misread it and thought "take the greatest factor" meant you were subtracting it from your original number before subtracting the other factor from that. I reread it like 3 times and never caught my mistake. Oops.

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7

Implosion has 14 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).

DeathRowKitty has 16 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has 14 points and:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [79, 87, 92] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9

popsofctown has 7 points and:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [4, 16, 25, 81] {
  n²
  } squares
  
• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite numbers such that its greatest non-trivial divisor minus one of its other divisors is a perfect number

StrangerCoug has 7 points and:

• [1, 10, 15, 28] {
  n*(n-1)/2)
  }: triangular numbers

There are 88 cards left in the deck. It is StrangerCoug's turn

[popsofctown]

I'm worried that that phrasing of my sequence might make people think that the "other divisor" must be non-trivial. 58 requires the use of a trivial divisor as the other divisor.

Maybe I'm being too worrisome though, a trivial divisor is indeed an "other divisor".

[Plotinus]

How do you feel about this wording "composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number"?

[Plotinus]

i think i mixed up my plurals/singulars in there somehow but i can't figure out how to untangle it

[implosion]

i mean, i'm forced to interpret "preferred factor" as a factor that lives in the sequence here :p

(and i think that definition looks good plot)

[StrangerCoug]

[13, 27, 72]: Numbers whose digital sum is a square

1 + 3 = 4, √4 = 2
2 + 7 = 7 + 2 = 9, √9 = 3

[StrangerCoug]

Oh, and since 1, 4, and 9 are one-digit squares, their digital sums are trivially squares, so I don't mind them in the above.

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)

McMenno has:

• [14, 35, 343] {
  7n
  } divisible by 7

Implosion has 14 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).

DeathRowKitty has 16 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has 14 points and:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [79, 87, 92] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9

popsofctown has 7 points and:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [4, 16, 25, 81] {
  n²
  } squares
  
• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number

StrangerCoug has 7 points and:

• [1, 10, 15, 28] {
  n*(n-1)/2)
  }: triangular numbers
  
• [13, 27, 72] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square

There are 85 cards left in the deck. It is McMenno's turn.

[StrangerCoug]

Mod: By post #110, my intended clarification was that one-digit squares are allowed in the sequence I just played; are you sure your closed form doesn't imply the contrary?

[Plotinus]

a_i can be 0, so 4 could be written as 0*1000 + 0*100 + 0*10 + 4*1, but i should edit it to show that a_i can't be negative.

How's this?

[StrangerCoug]

Looks good

[Plotinus]

McMenno has been prodded. It will be implosion's turn in (expired on 2019-10-23 19:16:00) or as soon as McMenno goes, whichever happens first.

This is McMenno's second prod in a row. That means that if McMenno doesn't go, he will be marked inactive and autoskipped until he tells us that he is back.

[Plotinus]

McMenno has been marked inactive. It is implosion's turn.

[implosion]

Numbers less than 50 with exactly 1 prime factor, *excluding 27 and 19*: 9, 37, 25, 7.

[popsofctown]

That doesn't seem within the spirit of the game

[implosion]

In what way?

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)

McMenno is inactive and has:

• [14, 35, 343] {
  7n
  } divisible by 7

Implosion has 14 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).
  
• [7, 9, 25, 37] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor

DeathRowKitty has 16 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has 14 points and:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [79, 87, 92] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9

popsofctown has 7 points and:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [4, 16, 25, 81] {
  n²
  } squares
  
• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number

StrangerCoug has 7 points and:

• [1, 10, 15, 28] {
  n*(n-1)/2)
  }: triangular numbers
  
• [13, 27, 72] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square

There are 81 cards left in the deck. It is DeathRowKitty's turn.

powers of primes usually includes 1, because p⁰ = 1 but the way you wrote it I think 1 is not included because by that logic 1 would have infinitely many prime factors, but let me know if I misunderstood.

[DeathRowKitty]

Add [36, 78, 120] to triangular numbers

[Plotinus]

Spoiler: Finished sequences:

• [5, 9, 10, 27, 48, 66, 98] {
  n² ± [0, 2]
  } numbers within 2 of a perfect square,
  
• [13, 15, 51, 53, 55, 73, 91] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with a_i ≡ 1 (mod 2) ∀ i ∈ [1, 9], ∀ d > 0
  } numbers with at least two digits, all of which are odd
  
• [10, 21, 23, 45, 46, 59, 70] {
  n is 0 or odd (mod 9)
  } numbers that are odd when you repeatedly sum their digits
  
• [6, 7, 8, 13, 18, 20, 24, 34, 40] n is the sum of the Scrabble point values of the letters in the US spelling of the numbers in the deck.
  
• [12, 20, 35, 62, 85, 95, 100] numbers that are the sum of the proper divisors of some number < 1000 not in the deck for this game.
  
• [11, 17, 19, 29, 43, 71, 83] primes
  
• [5, 6, 13, 15, 16, 27, 100] numbers used in 0, including substrings of other numbers, but not including the deck spoiler
  
• [10, 21, 56, 57, 64, 76, 729] integers n for which there exists some integer m such that (n-1)/3^m and (n-2)/3^m are each endpoints of intervals removed during (possibly different) steps of the usual construction of the Cantor set (i.e. the construction in which each step removes the middle third of intervals existing after the previous step)
  
• [1, 10, 15, 28, 36, 78, 120] {
  n*(n-1)/2)
  }: triangular numbers

McMenno is inactive and has:

• [14, 35, 343] {
  7n
  } divisible by 7

Implosion has 14 points and:

• [30, 42, 65] Composite squarefree numbers where when you take the sum of prime factors and write it in english, at least 1/3 of the letters in the word are "e"
  
• [9, 17, 69, 77] {
  n ≡ 1 (mod 4) ∧ n ≥ 7 (mod 10)
  } Numbers congruent to 1 mod 4 whose last digit, written in english, can have the letters "ty" appended to the end of it to multiply it by ten (e.g., "six" times ten is "sixty", but "fourty" is not a number, so the last digit cannot be four)
  
• [5, 6, 50, 125] Numbers such that if you take the number of letters in the english spelling and add that to the number, and then repeat that process a second time, the result is in the range 11-15 mod 50 (inclusive).
  
• [7, 9, 25, 37] {
  p^k | p^k < 50, prime p, k > 0
  }
  \
  {
  19, 27
  } numbers less than 50 with exactly 1 prime factor

DeathRowKitty has 23 points and:

• [38, 82, 84] slots never touched by Ace, 5, or 9 in perfect out-shuffles of standard 52 card decks, mod 52

Felissan has 14 points and:

• [2, 4, 32, 256] {
  2ⁿ
  } powers of two
  
• [30, 40, 55] {
  25 + (5n * (n + 1) / 2)
  } 25 + 5n, where n is a triangular number
  
• [79, 87, 92] {
  maxdigit(n) > 7
  } numbers that contain an 8 or 9

popsofctown has 7 points and:

• [4, 20, 36, 68] {
  16n + 4
  } remainder is 4 when dividing by 16
  
• [4, 16, 25, 81] {
  n²
  } squares
  
• [55, 58, 60] {
  n, k, c st n is composite; k - c is perfect; c|n; k = max(d) st d|n ∧d ≠ n
  } composite number whose greatest non-trivial divisor minus any of its other divisors is a perfect number

StrangerCoug has 7 points and:

• [13, 27, 72] {
  n = a₀10^d + a₁10^d-1 + ... + a_d10⁰ with Σ_i∈[0,d]a_i = k^2 | a_i ≥ 0, d > 0, k ∈ ℤ
  } numbers whose digit sum is a square

There are 78 cards left in the deck. It is Felissan's turn.

Do you guys want me to replace McMenno or do you want to continue with 5 players until he reappears? I know someone who was interested when popsofctown replaced in.

[implosion]

powers of primes usually includes 1, because p0 = 1 but the way you wrote it I think 1 is not included because by that logic 1 would have infinitely many prime factors, but let me know if I misunderstood.

Yeah, the way you wrote it is good. My intent was 1 isn't included because it has 0 prime factors.

[popsofctown]

Excluding 19 and 27 from the sequence is lame. Something somewhat natural like perfect cubes but excluding perfect squares would be fine. Excluding two numbers because they aren't in your hand or for no reason at all seems unfun.