Fausto A. C. Cariboni - cp's OEIS frontend

A344635 Number of knapsack partitions of n with largest part 10.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 7, 11, 12, 17, 1, 13, 9, 16, 11, 20, 14, 24, 16, 25, 9, 27, 14, 29, 19, 32, 16, 34, 19, 37, 11, 32, 17, 38, 19, 32, 22, 41, 19, 40, 14, 38, 22, 41, 22, 39, 18, 44, 26, 46, 8, 46, 24, 38, 23, 40, 21, 48, 28, 42, 12

Offset: 0

Fausto A. C. Cariboni, May 25 2021

nonn

An integer partition is knapsack if every distinct submultiset has a different sum.

I computed terms a(n) for n = 0..50000 and the subsequence a(162)-a(2681) of length 2520 is repeated continuously.

			The initial nonzero values count the following partitions:
  10: (10)
  11: (10,1)
  12: (10,1,1), (10,2)
  13: (10,1,1,1), (10,2,1), (10,3)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..3000

Cf. A108917, A275972, A326017, A326034, A342684, A343321, A344310, A344340, A344412, A344625.

A344625 Number of knapsack partitions of n with largest part 9.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 7, 11, 12, 1, 10, 7, 11, 10, 17, 12, 18, 16, 12, 15, 19, 13, 25, 20, 17, 22, 29, 6, 25, 20, 22, 20, 28, 16, 31, 21, 14, 23, 33, 15, 24, 22, 25, 28, 30, 8, 31, 20, 22, 22, 36, 16, 34, 26, 14, 23, 26, 22, 33, 25, 24

Offset: 0

Fausto A. C. Cariboni, May 25 2021

nonn

An integer partition is knapsack if every distinct submultiset has a different sum.

I computed terms a(n) for n = 0..50000 and the subsequence a(128)-a(2647) of length 2520 is repeated continuously.

			The initial nonzero values count the following partitions:
   9: (9)
  10: (9,1)
  11: (9,1,1), (9,2)
  12: (9,1,1,1), (9,2,1), (9,3)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..3000

Cf. A108917, A275972, A326017, A326034, A342684, A343321, A344310, A344340, A344412.

A342684 Number of knapsack partitions of n with largest part 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 7, 11, 1, 8, 6, 10, 7, 13, 9, 15, 6, 12, 10, 15, 8, 18, 10, 17, 6, 17, 12, 17, 9, 18, 13, 22, 7, 19, 10, 19, 13, 20, 14, 24, 4, 20, 12, 19, 13, 23, 15, 21, 4, 20, 13, 23, 11, 23, 15, 20, 7, 20, 12, 22, 15, 24, 12, 22

Offset: 0

Fausto A. C. Cariboni, May 18 2021

nonn

An integer partition is knapsack if every distinct submultiset has a different sum.

I computed terms a(n) for n = 0..40000 and the subsequence a(98)-a(937) of length 840 is repeated continuously.

			The initial nonzero values count the following partitions:
   8: (8)
   9: (8,1)
  10: (8,1,1), (8,2)
  11: (8,1,1,1), (8,2,1), (8,3)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..1000

Cf. A108917, A275972, A326034, A343321, A344310, A344340, A344412.

A344412 Number of knapsack partitions of n with largest part 7.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 7, 1, 6, 5, 8, 7, 10, 8, 8, 9, 11, 8, 13, 11, 13, 5, 14, 8, 13, 10, 17, 12, 8, 10, 14, 13, 14, 12, 18, 3, 15, 11, 15, 14, 17, 12, 8, 12, 15, 13, 20, 12, 14, 5, 17, 15, 17, 10, 18, 14, 9, 13, 18, 13, 15, 15, 18, 5, 18, 11

Offset: 0

Fausto A. C. Cariboni, May 17 2021

nonn

An integer partition is knapsack if every distinct submultiset has a different sum.

I computed terms a(n) for n = 0..25000 and the subsequence a(72)-a(491) of length 420 is repeated continuously.

			The initial nonzero values count the following partitions:
   7: (7)
   8: (7,1)
   9: (7,1,1), (7,2)
  10: (7,1,1,1), (7,2,1), (7,3)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..1000

Cf. A108917, A275972, A326017, A326034, A343321, A344310, A344340.

A344340 Number of knapsack partitions of n with largest part 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 1, 4, 4, 6, 5, 7, 3, 7, 4, 8, 6, 10, 2, 7, 6, 9, 6, 9, 2, 9, 5, 9, 7, 9, 2, 8, 7, 10, 5, 9, 3, 10, 6, 8, 7, 10, 3, 9, 6, 10, 6, 10, 4, 9, 6, 9, 8, 11, 1, 9, 7, 11, 7, 8, 3, 10, 7, 10, 6, 10, 2, 10, 8, 9, 6, 9, 4, 11, 5, 9, 7

Offset: 0

Fausto A. C. Cariboni, May 15 2021

nonn

An integer partition is knapsack if every distinct submultiset has a different sum.

I computed terms a(n) for n = 0..10000 and (6,10,6,10,4,9,6,9,8,11,1,9,7,11,7,8,3,10,7,10,6,10,2,10,8,9,6,9,4,11,5,9,7,11,3,8,7,10,7,10,2,10,6,10,8,9,2,9,8,11,5,9,3,11,7,8,7,10,3,10) is repeated continuously starting at a(50).

			The initial values count the following partitions:
   6: (6)
   7: (6,1)
   8: (6,1,1)
   8: (6,2)
   9: (6,1,1,1)
   9: (6,2,1)
   9: (6,3)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..1000

Cf. A108917, A275972, A326017, A326034, A343321, A344310.

A343321 Number of knapsack partitions of n with largest part 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 1, 4, 3, 5, 5, 4, 4, 6, 5, 7, 2, 6, 5, 8, 5, 4, 6, 7, 6, 8, 2, 8, 6, 7, 7, 5, 5, 8, 7, 8, 2, 8, 6, 9, 6, 3, 7, 9, 5, 8, 3, 8, 6, 8, 6, 5, 6, 7, 7, 9, 1, 8, 7, 8, 6, 4, 6, 9, 6, 7, 3, 9, 5, 8, 7, 4, 6, 8, 6, 9, 2, 7, 7, 9, 5, 4, 7

Offset: 0

Fausto A. C. Cariboni, May 14 2021

nonn

An integer partition is knapsack if every distinct submultiset has a different sum.

I computed terms a(n) for n = 0..10000 and (6,7,7,5,5,8,7,8,2,8,6,9,6,3,7,9,5,8,3,8,6,8,6,5,6,7,7,9,1,8,7,8,6,4,6,9,6,7,3,9,5,8,7,4,6,8,6,9,2,7,7,9,5,4,7,8,6,8,2,9) is repeated continuously starting at a(32).

			The initial values count the following partitions:
   5: (5)
   6: (5,1)
   7: (5,1,1)
   7: (5,2)
   8: (5,1,1,1)
   8: (5,2,1)
   8: (5,3)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..1000

Cf. A108917, A275972, A326017, A326034, A344310.

A344310 Number of knapsack partitions of n with largest part 4.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 1, 2, 3, 3, 1, 3, 3, 3, 2, 3, 3, 4, 2, 3, 4, 4, 1, 4, 4, 3, 2, 4, 3, 4, 2, 3, 4, 4, 1, 4, 4, 3, 2, 4, 3, 4, 2, 3, 4, 4, 1, 4, 4, 3, 2, 4, 3, 4, 2, 3, 4, 4, 1, 4, 4, 3, 2, 4, 3, 4, 2, 3, 4, 4, 1, 4, 4, 3, 2, 4, 3, 4, 2, 3, 4, 4, 1, 4, 4

Offset: 0

Fausto A. C. Cariboni, May 14 2021

nonn

An integer partition is knapsack if every distinct submultiset has a different sum.

I computed terms a(n) for n = 0..10000 and (3, 4, 2, 3, 4, 4, 1, 4, 4, 3, 2, 4) is repeated continuously starting at a(18).

			The initial values count the following partitions:
   4: (4)
   5: (4,1)
   6: (4,1,1)
   6: (4,2)
   7: (4,1,1,1)
   7: (4,2,1)
   7: (4,3)
   8: (4,4)

Cf. A108917, A275972, A326017, A326034, A343321.

A340267 Maximum LCM of partitions of n into pairwise coprime parts that are >= 2.

Original entry on oeis.org

2, 3, 4, 6, 6, 12, 15, 20, 30, 30, 60, 42, 84, 105, 140, 210, 210, 420, 280, 330, 360, 840, 504, 1260, 1155, 1540, 2310, 2520, 4620, 3080, 5460, 3960, 9240, 5544, 13860, 6930, 16380, 15015, 27720, 30030, 32760, 60060, 40040, 45045, 51480, 120120, 72072, 180180

Offset: 2

Fausto A. C. Cariboni, Jan 02 2021

nonn

a(n) <= A123131(n).

			For n=22 we have a(22) = 360 since 22 = 5 + 8 + 9 and lcm([5, 8, 9]) = 360.
Note a(22) = 360 < A123131(22) = 420.

Fausto A. C. Cariboni, Table of n, a(n) for n = 2..260

Cf. A007359, A007360, A123131.

isok(p) = {for (i=1, #p, for (j=i+1, #p, if (gcd(p[i], p[j]) > 1, return(0)););); return(1);}
a(n) = {my(x=1); forpart(p=n, if ((vecmin(p)>=2) && isok(p), x = max(x, lcm(Vec(p))));); x;} \\ Michel Marcus, Jan 03 2021

A338737 Triangle read by rows: T(n,k) is the number of sum-free subsets of {1..n} with cardinality k for 1 <= k <= n.

Original entry on oeis.org

1, 2, 0, 3, 2, 0, 4, 4, 0, 0, 5, 8, 2, 0, 0, 6, 12, 5, 0, 0, 0, 7, 18, 14, 2, 0, 0, 0, 8, 24, 24, 4, 0, 0, 0, 0, 9, 32, 45, 19, 2, 0, 0, 0, 0, 10, 40, 65, 32, 3, 0, 0, 0, 0, 0, 11, 50, 100, 72, 17, 2, 0, 0, 0, 0, 0, 12, 60, 137, 121, 35, 3, 0, 0, 0, 0, 0, 0

Offset: 1

Fausto A. C. Cariboni, Nov 05 2020

			The 8 sum-free subsets of {1,2,3,4} with at least one element are {1}, {2}, {3}, {4}, {1,3}, {1,4}, {2,3}, {3,4}, hence the 4th row is 4,4,0,0.
The triangle begins:
   1;
   2,  0;
   3,  2,  0;
   4,  4,  0,  0;
   5,  8,  2,  0,  0;
   ...

Fausto A. C. Cariboni, Rows n = 1..70, flattened

Cf. A007865, A288887, A288888.

sumfree(v) = {for(i=1, #v, for (j=1, i, if (setsearch(v, v[i]+v[j]), return (0)););); return (1);}
row(n) = {my(v = vector(n)); forsubset(n, s, if (#s && sumfree(Set(s)), v[#s]++);); v;} \\ Michel Marcus, Nov 08 2020

A338564 Number of cyclic arrangements of {1..n} such that any three neighbors satisfy the triangle inequality.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 2, 17, 51, 175, 827, 3951, 20428, 115262, 692851, 4324011, 29446302, 211048631, 1623253741, 13109205113

Offset: 3

Fausto A. C. Cariboni, Nov 02 2020

Reversals of these circular permutations are not counted as different.

Fausto A. C. Cariboni, Complete solutions for a(3)-a(11)

Cf. A336351.

cp's OEIS Frontend

User: Fausto A. C. Cariboni

A344635 Number of knapsack partitions of n with largest part 10.

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A344625 Number of knapsack partitions of n with largest part 9.

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A342684 Number of knapsack partitions of n with largest part 8.

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A344412 Number of knapsack partitions of n with largest part 7.

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A344340 Number of knapsack partitions of n with largest part 6.

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A343321 Number of knapsack partitions of n with largest part 5.

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A344310 Number of knapsack partitions of n with largest part 4.

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A340267 Maximum LCM of partitions of n into pairwise coprime parts that are >= 2.

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A338737 Triangle read by rows: T(n,k) is the number of sum-free subsets of {1..n} with cardinality k for 1 <= k <= n.

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Examples

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Programs

PARI

A338564 Number of cyclic arrangements of {1..n} such that any three neighbors satisfy the triangle inequality.

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