A142153 -id:A142153 - cp's OEIS frontend

A141494 a(n) is the "highest smallest" positive integer that cannot be obtained from the (n-1) optimized integers (to be defined for each n) using each number at most once and the three operators +, -, *.

1, 2, 5, 18, 70, 406

Offset: 1

Gilles A.Fleury, Aug 10 2008, Aug 24 2008; revised Oct 05 2008

This sequence is a kind of optimized version of the sequence A060315 for which the inputs are the integers {0,1,...,n-1}. Here the inputs are optimized so that the smallest positive integer, that cannot be obtained, is maximized.

Further terms may be hard to find. Some additional terms (still to be proved) could be a(7)=2876, a(8)=24756, a(9)=346404. If anyone has found higher numbers please contact me.

			a(4)=18 because every integer can be calculated up to 17, using the optimal numbers {2,3,10}.
a(5)=70 because every integer can be calculated up to 69, using one of the two (!) optimal sequences {2,3,4,27} or {2,3,10,41}.
a(6)=406 because every integer can be calculated up to 405, using the optimal numbers {2,3,4,84,111}.

Cf. A060315, A142153.

A143191 a(n) is the smallest natural number we cannot obtain from n, n+1, n+2, n+3, n+4, n+5, n+6 and the operators +, -, *, /, using each number only once.

Original entry on oeis.org

284, 1413, 2113, 3266, 4943, 6242, 9105, 11586, 6269, 6427, 8407, 8406, 9224, 11079, 12451, 8392, 3469, 4253, 4043, 4126, 4087, 4657, 4330, 4639, 5114, 3983, 5839, 4415, 6376, 4537, 5231, 5161, 4090, 3199, 2057, 3372, 2285, 2270, 2525, 2609, 2590, 1209

Offset: 0

Gilles A.Fleury, Oct 18 2008

nonn

This sequence is related to the sequences A071110 (for 5 successive integers) and A060316 (for 6 successive integers) and others sequences to come...

Asymptotically, the sequence tends to 29 (the first n for which a(n)=29 is n=249).

Gilles A. Fleury, Table of n, a(n) for n=0..300

Cf. A071110, A060316, A071107, A060315, A141494, A142153.

A143192 a(n) is the smallest natural number we cannot obtain from n, n+1, n+2, n+3, n+4, n+5, n+6, n+7 and the operators +, -, *, /, using each number only once.

Original entry on oeis.org

1413, 7187, 12421, 22751, 28862, 48046, 36094, 46372, 54214, 72845, 88119, 107246, 125589, 104153, 43838, 45893, 55054, 62090, 66226, 70187, 69638, 74941, 85303, 81913, 68891, 77237, 37997, 48758, 42827, 45554, 22217, 26617, 29422, 29099

Offset: 0

Gilles A.Fleury, Oct 18 2008, Mar 06 2009

nonn

This sequence is related to the sequences A071110 (for 5 successive integers) and A060316 (for 6 successive integers) and others sequences to come...

Asymptotically the sequence tends to 67 (the first n for which a(n)=67 is n=1042).

Gilles A. Fleury, Table of n, a(n) for n = 0..1500

Cf. A071110, A060316, A071107, A060315, A141494, A142153.

A143193 a(n) is the smallest natural number we cannot obtain from n, n+1, n+2, n+3, n+4, n+5, n+6, n+7, n+8 and the operators +, -, *, /, using each number only once.

Original entry on oeis.org

7187, 38103, 54251, 114358, 168673, 264111, 319699, 456061, 588847, 812092, 1005321, 1222630, 445059, 499063, 600907, 706847, 820609, 929113, 1048137, 1269847, 1049291, 1113439, 1252843, 1411942, 1588841, 456206, 462382, 464357, 479894

Offset: 1

Gilles A.Fleury, Oct 18 2008

nonn

This sequence is related to the sequences A071110 (for 5 successive integers) and A060316 (for 6 successive integers).

What is the asymptotic value of this sequence? What is the first n for which a(n) equals the asymptotic value?

Gilles A. Fleury, Table of n, a(n) for n = 1..51

Cf. A071110, A060316, A071107, A060315, A141494, A142153.

A309885 a(n) is the largest integer k such that there exists a set of n integers from which each number in 1,...,k can be built using the basic operations +,-,*,/, with parentheses allowed, and using each element of the set exactly once.

Original entry on oeis.org

1, 3, 10, 52

Offset: 1

Matej Veselovac, Aug 21 2019

The next term, a(5), is at least 351. Is it true a(5)=351? Proving a value is a term of the sequence is hard. Extensive computations suggest a(5)=351 should be the next term.
Can we find lower bounds for a(6), a(7), ...? For example, a(6) >= 2200 which can be obtained by using the set of integers {2, 10, 13, 30, 49, 56}. Can we find better sets for n=6 ?
By "set of n integers" it is meant that a fixed set of exactly n not necessarily distinct positive integers is used for obtaining every number from 1 to a(n).
This sequence is similar to A142153, but there not all integers from the set must be used when building numbers, and here we are taking a(n) to be the "last obtainable number", instead of the "highest smallest unobtainable" number which is one more than the last obtainable number.
We can use A142153(n+1)-1 > a(n) for a upper bound, for n > 1. For a lower bound, we can use a(n) >= A309886(n)-1, where the inequality is strict for n > 3.
For a trivial, not sharp, upper bound, we can count the possible expressions that can be built with n digits and allowed operations, and find (taking into consideration +,* are commutative): A221954(n) > a(n), for n > 1.

			a(1) = 1 is trivial since binary operations *,+,-,/ are not applicable.
a(2) = 3 since we can make 1,2,3 but not 4, using the number set {1,2}.
a(3) = 10 since we can make 1,...,10 but not 11, using the number set {1,2,4}.
a(4) = 52 since we can make 1,...,52 but not 53, using the number set {2,3,4,22}.
a(5) >= 351 since we can make first 351 numbers using the number set {2,3,6,12,37}.

Math StackExchange, Largest consecutive integer using basic operations and optimal digits?

Cf. A142153, A309886.

Cf. A221954.

A143190 a(n) is the smallest natural number we cannot obtain from n, n+1, n+2, n+3 and the operators +, -, *, /, using each number only once.

Original entry on oeis.org

10, 29, 41, 43, 40, 44, 26, 21, 15, 15, 18, 18, 18, 10, 10, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Gilles A.Fleury, Oct 18 2008

nonn

This sequence is related to the sequences A071110 (for 5 successive integers) and A060316 (for 6 successive integers) and others sequences to come...

Asymptotically (in fact as soon as n>=15), the sequence tends to 5.

Cf. A071110, A060316, A071107, A060315, A141494, A142153.

cp's OEIS Frontend

A141494 a(n) is the "highest smallest" positive integer that cannot be obtained from the (n-1) optimized integers (to be defined for each n) using each number at most once and the three operators +, -, *.

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A143191 a(n) is the smallest natural number we cannot obtain from n, n+1, n+2, n+3, n+4, n+5, n+6 and the operators +, -, *, /, using each number only once.

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A143192 a(n) is the smallest natural number we cannot obtain from n, n+1, n+2, n+3, n+4, n+5, n+6, n+7 and the operators +, -, *, /, using each number only once.

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A143193 a(n) is the smallest natural number we cannot obtain from n, n+1, n+2, n+3, n+4, n+5, n+6, n+7, n+8 and the operators +, -, *, /, using each number only once.

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A309885 a(n) is the largest integer k such that there exists a set of n integers from which each number in 1,...,k can be built using the basic operations +,-,*,/, with parentheses allowed, and using each element of the set exactly once.

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A143190 a(n) is the smallest natural number we cannot obtain from n, n+1, n+2, n+3 and the operators +, -, *, /, using each number only once.

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