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User: Jake Bird

Jake Bird has authored 3 sequences.

A378759 Number of 1's required to build n using +, /, and ^.

1, 2, 3, 4, 5, 6, 7, 5, 5, 6, 7, 8, 9, 9, 10, 6, 7, 8, 9, 10, 11, 12, 13, 11, 7, 8, 6, 7, 8, 9, 10, 7, 8, 9, 10, 8, 9, 10, 11, 12, 10, 11, 12, 13, 13, 14, 15, 13, 9, 10, 11, 12, 13, 12, 13, 14, 13, 14, 13, 14, 12, 13, 11, 7, 8, 9, 10, 11, 12, 13, 14, 12

Offset: 1

Jake Bird, Dec 06 2024

nonn

All intermediate steps in building the number should also be integers.

A348262(n) >= a(n) >= A348089(n) for all n, as the available operators in A348262 are a subset of the available operators here, and the available operators here are a subset of the available operators in A348089.

			a(14) = 9 because 14 = ((1+1+1)^(1+1+1)+1)/(1+1), which has 9 occurrences of the symbol "1", and there is no way of making 14 with fewer using these rules.
Note that A348262(14) = 10 because 14 = (1+1+1)^(1+1)+(1+1)^(1+1)+1; division allows for one fewer occurrence of the symbol "1" to be used here. Similarly, A348089(14) = 8, because 14 = (1+1)^(1+1)^(1+1)-(1+1); subtraction allows for one fewer occurrence of the symbol "1" to be used there. 14 is the least n such that A348262(n) > a(n) > A348089(n).

Jake Bird, Table of n, a(n) for n = 1..119

Cf. A000027 {1,+}, {1,+,-}
Cf. A005245 {1,+,*}
Cf. A348262 {1,+,^}
Cf. A091333 {1,+,-,*}
Cf. A378758 {1,+,-,^}
Cf. A025280 {1,+,*,^}
Cf. A091334 {1,+,-,*,^}
Cf. A348089 {1,+,-,*,/,^}

A378758 Number of 1's required to build n using +, -, and ^.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 6, 5, 5, 6, 7, 8, 9, 8, 7, 6, 7, 8, 9, 10, 11, 10, 9, 8, 7, 7, 6, 7, 8, 9, 8, 7, 8, 9, 9, 8, 9, 10, 11, 12, 12, 13, 12, 13, 12, 11, 10, 9, 8, 9, 10, 11, 12, 12, 12, 12, 13, 13, 12, 11, 10, 9, 8, 7, 8, 9, 10, 11, 12, 13, 13, 12, 12, 13

Offset: 1

Jake Bird, Dec 06 2024

nonn

All intermediate steps in building the number should be integers as well, for consistency with related sequences.

A348262(n) >= a(n) >= A091334(n) for all n, as the available operators in A348262 are a subset of the available operators here, and the available operators here are a subset of the available operators in A091334.

			a(22) = 10 because 22 = (1+1+1+1+1)^(1+1)-(1+1+1), which has 10 occurrences of the symbol "1", and there is no way of making 22 with fewer using these rules.
Note that A348262(22) = 12 because 22 = (1+1)^(1+1)^(1+1)+(1+1)^(1+1)+1+1; subtraction allows for two fewer occurrences of the symbol "1" to be used here. Similarly, A091334(22) = 9 because 22 = ((1+1+1)^(1+1)+1+1)*(1+1); multiplication allows for one fewer occurrence of the symbol "1" to be used there. 22 is the least n such that A348262(n) > a(n) > A091334(n).

Jake Bird, Table of n, a(n) for n = 1..300

Cf. A000027 {1,+}, {1,+,-}
Cf. A005245 {1,+,*}
Cf. A348262 {1,+,^}
Cf. A091333 {1,+,-,*}
Cf. A025280 {1,+,*,^}
Cf. A378759 {1,+,/,^}
Cf. A091334 {1,+,-,*,^}
Cf. A348089 {1,+,-,*,/,^}

A376150 Define b_n(k) to be the lexicographically earliest sequence of distinct nonnegative integers with the property that two terms that contain the digit "d" are always separated by exactly "d" terms that do not contain the digit "d", in base n. a(n) is the number of terms in b_n(k).

Original entry on oeis.org

2, 4, 6, 10, 10, 18, 18, 22, 22, 30, 30, 34, 42, 42, 78, 78, 78, 78, 102, 102, 114, 114, 114, 114, 142, 142, 142, 142, 214, 214, 214, 214, 214, 214, 214, 222, 274, 274, 274, 274, 274, 354, 354, 354, 354, 354, 354, 642, 642, 642, 642, 642, 642, 642, 642

Offset: 2

Jake Bird, Sep 12 2024

This process terminates only when all nonzero digits are prohibited by the restrictions in place for the next term; as b_n(2) = "10" for all n, the digit 1 is only prohibited for odd numbered terms, and as such a(n) must be even for all n. Similar logic can be applied to the digit 3 to show that for all n>3, a(n) is not divisible by 4.

A375232 is the sequence generated when n=10.

			For n = 5:
b_5(1) = 0; as this contains the digit 0, b_5(2), b_5(3) etc. must also contain a 0
b_5(2) = 10 (= 5 in decimal); must contain a 0 from b_5(1); as this contains the digit 1, b_5(4), b_5(6) etc. must also contain a 1, and all other terms must not contain a 1
b_5(3) = 20; must have 0 but not 1
b_5(4) = 100; must have 0 and 1 but not 2
b_5(5) = 30; must have 0 but not 1 or 2
b_5(6) = 102; must have 0, 1, and 2, but not 3
b_5(7) = 40; must have 0 but not 1, 2, or 3
b_5(8) = 101; must have 0 and 1 but not 2, 3, or 4
b_5(9) = 203; must have 0, 2, and 3, but not 1 or 4
b_5(10) = 110; must have 0 and 1 but not 2, 3, or 4
b_5(11) = ---; must have 0 but not 1, 2, 3, or 4 - the only number that fills this condition is 0, but 0 already appears in the sequence, so the sequence terminates after ten terms, and a(5) = 10

Jake Bird, Table of n, a(n) for n = 2..88

Cf. A375232.

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User: Jake Bird

A378759 Number of 1's required to build n using +, /, and ^.

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A378758 Number of 1's required to build n using +, -, and ^.

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A376150 Define b_n(k) to be the lexicographically earliest sequence of distinct nonnegative integers with the property that two terms that contain the digit "d" are always separated by exactly "d" terms that do not contain the digit "d", in base n. a(n) is the number of terms in b_n(k).

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