A141494 -id:A141494 - cp's OEIS frontend

A060315 a(1)=1; a(n) is the smallest positive integer that cannot be obtained from the integers {0, 1, ..., n-1} using each number at most once and the operators +, -, *, /.

1, 2, 4, 10, 29, 76, 284, 1413, 7187, 38103, 231051, 1765186, 10539427

Offset: 1

Jean-Marc Rebert, Mar 28 2001

I had written a C++ program to find the smallest positive integer that cannot be obtained from the integers {1,2,...,n-1} using each number exactly once and the operators +,-,*,/. The result is the same as this sequence through n=11. It takes the program two days to find the result for n=11. We still don't know whether the two sequences are the same for n greater than 11. - Zhao Hui Du, Oct 01 2008
The first 12 terms are the same as the result of using all numbers from 0 to n-1 exactly once and only the operators +,-,* (so we could get all integers less than a(n) without the operator /). The minimal number which could not be reached using all numbers from 0 to 12 exactly once and only operators +,-,* is 10539427. But I have still not verified whether it is a(13). - Zhao Hui Du, Oct 08 2008
a(13) has now been verified by computer. - Zhao Hui Du, Nov 05 2008

			For n=4 the numbers available for use are {0,1,2,3} and we can get 6=2*3, 7=2*3+1, 8=2*(1+3), 9=3*(1+2), but we cannot get 10, hence a(4) = 10.

Gilles Bannay, Countdown Problem (to obtain a(4)=10 for example, enter ceb -a4 -x1 0 1 2 3)
Michael S. Branicky, Python program for OEIS A060315 and its +,-,* version
Zhao Hui Du, The C++ source code to find the smallest integer [The old link to the program was broken, but _Zhao Hui Du_ kindly sent this email with a copy of the code. - _N. J. A. Sloane_, Jul 04 2022]
Zhao Hui Du, The webpage where the result is posted [From _Zhao Hui Du_, Oct 08 2008]
Index entries for similar sequences

Cf. A060316, A141494, A354423.

a(n) >= A354423(n-1). - Michael S. Branicky, Jun 05 2022

More terms from Koksal Karakus (karakusk(AT)hotmail.com), May 26 2002
One more term from Zhao Hui Du, Oct 08 2008
Replaced two broken links with a link to a local copy of the missing program. - N. J. A. Sloane, Jul 04 2022

A060316 a(n) is the smallest natural number we cannot obtain from n, n+1, n+2, n+3, n+4, n+5 and the operators +, -, *, /, using each number only once.

Original entry on oeis.org

76, 284, 433, 734, 842, 1102, 1228, 1366, 1652, 709, 859, 879, 943, 1070, 1091, 749, 829, 530, 628, 653, 677, 202, 342, 293, 248, 238, 247, 245, 253, 336, 251, 147, 125, 155, 127, 163, 139, 133, 149, 181, 157, 153, 155, 169, 162, 157, 131, 174, 176, 169

Offset: 0

Jean-Marc Rebert, Mar 28 2001

nonn

Asymptotically the sequence tends to 13. The first n for which a(n) equals the limit is n=83. - Gilles A.Fleury, Oct 18 2008

			u(0)=76 is the smallest natural number we can't obtain with 0, 1, 2, 3, 4, 5 and the operators +, -, *, /, using each number only once.

Gilles A. Fleury, Table of n, a(n) for n = 0..100, Oct 18 2008.
Gilles Bannay, Countdown Problem
Index entries for similar sequences

Cf. A060315, A141494.

More terms from Koksal Karakus (karakusk(AT)hotmail.com), May 28 2002

A142153 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 operators +, -, *, /.

Original entry on oeis.org

1, 2, 5, 18, 87, 451

Offset: 1

Gilles A.Fleury, 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)=3495, a(8)=32355, a(9)=384289. If anyone has found higher numbers please contact me. - updated by Gilles A.Fleury, Jul 10 2017 and May 22 2018

			a(4) = 18 because every integer can be calculated up to 17, using one of the four (!) optimal sequences {2,3,10} or {2,3,14} or {2,6,11} or {2,6,13}.
a(5) = 87 because every integer can be calculated up to 86, using the optimal numbers {2,3,14,60}.
a(6) = 451 because every integer can be calculated up to 450, using the optimal numbers {2,3,4,63,152}. - _Gilles A.Fleury_, Mar 06 2009

Cf. A141494, A060315.

a(6) from Gilles A.Fleury, Mar 06 2009

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.

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

A060315 a(1)=1; a(n) is the smallest positive integer that cannot be obtained from the integers {0, 1, ..., n-1} using each number at most once and the operators +, -, *, /.

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A060316 a(n) is the smallest natural number we cannot obtain from n, n+1, n+2, n+3, n+4, n+5 and the operators +, -, *, /, using each number only once.

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A142153 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 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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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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