A135415 -id:A135415 - cp's OEIS frontend

A100480 a(1)=1 and, for n>1, a(n) is the smallest positive integer such that the subset S(c) of {1,2,3,...,n} defined by S(c)={k|1<=k<=n; a(k)=c} does not contain a 3-term arithmetic progression for any integer c.

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

Offset: 1

John W. Layman, Nov 22 2004

nonn

The sequence of values of n for which a(n)=1 is given by {A005836(i)-1}.

Samuel B. Reid, Table of n, a(n) for n = 1..10000
Samuel B. Reid, Python program for A100480

Cf. A005836, A006997, A135415, A229037.

Python
```
# See Links section.
```

a(n + 1) = A006997(n) + 1 for any n >= 0. - Rémy Sigrist, Jun 28 2022

A157102 Tuple-chromatic Van der Waerden numbers.

Original entry on oeis.org

3, 7, 7, 21, 11, 43, 15, 25, 19, 111, 23, 157, 27, 43, 31, 273, 35, 343, 39, 61, 43, 507, 47, 121, 51, 79, 55, 813, 59, 931, 63, 97, 67, 171, 71, 1333, 75, 115, 79, 1641, 83, 1807, 87, 133, 91, 2163, 95, 337, 99, 151, 103, 2757, 107, 271, 111, 169, 115, 3423, 119, 3661

Offset: 2

Reed Kelly, Feb 22 2009, Feb 25 2009

nonn

See links for definition. Specifically, the terms of this sequence are the first several terms of tcW(r,r-1,r), where r=2,3,4,.... Informally, the function tcW is like the multi-color Van der Waerden function W, except that the second parameter determines the number of colors found in the target subsequence. If W(r,k) is the standard multi-color Van der Waerden function with r colors and a required monochrome arithmetic subsequence of length k, then tcW(r,1,k) = W(r,k). In tcW(r,1,k), the 1 would indicate a monochrome subsequence. For tcW(r,2,k) an arithmetic subsequence of length k in 1 OR 2 colors would match the criteria. For tcW(r,3,k) an arithmetic subsequence of length k in 1, 2, or 3 colors suffices.

a(r) = tcW(r,r-1,r).

			a(2) = tcW(2,1,2) = W(2,2) = 3. If {1,2,3} is colored in 2 colors, then a 2 term arithmetic subsequence exists in 1 color (monochrome).
a(3) = tcW(3,2,3) = 7. If {1,...,7} is colored in 3 colors, then a 3 term arithmetic subsequence exists that is colored in at most 2 colors.
a(2) = (2-1)(2) + 1 = 3 a(15) = (15-1)(3) + 1 = 43.

Reed Kelly, On a Generalization of Ramsey Theory, 2009.
Reed Kelly, Code for Computing Tuple-chromatic Ramsey Numbers and Tuple-chromatic Van der Waerden Numbers

The 2-color Van der Waerden numbers: A005346, W(2, k). Multi-color Van der Waerden numbers with 3 term monochrome arithmetic subsequences A135415, W(r, 3).

Table[(x - 1) * (FactorInteger[x])[[1]][[1]] + 1, {x, 2, 100}]

a(n) = (n-1)*(smallest prime factor of n) + 1.

A157199 The terms of this sequence are the first several terms of tcW(r,r-1,r+1), where r=2,3,4,.... Informally, the function tcW is like the multi-color Van der Waerden function W, except that the second parameter determines the number of colors found in the target subsequence. See links for definition.

Original entry on oeis.org

9, 13, 22, 26, 44, 50, 25, 28

Offset: 2

Reed Kelly, Feb 25 2009

If W(r,k) is the standard multi-color Van der Waerden function with r colors and a required monochrome arithmetic subsequence of length k, then tcW(r,1,k) = W(r,k). In tcW(r,1,k), the 1 would indicate a monochrome subsequence. For tcW(r,2,k) an arithmetic subsequence of length k in 1 OR 2 colors would match the criteria. For tcW(r,3,k) an arithmetic subsequence of length k in 1, 2, or 3 colors suffices.

			a(2) = tcW(2,1,3) = W(2,3) = 9. If {1,...,9} is colored in 2 colors, then a 3-term arithmetic subsequence exists in 1 color (monochrome).
a(3) = tcW(3,2,4) = 13. If {1,...,13} is colored in 3 colors, then a 4-term arithmetic subsequence exists in at most 2 colors.

Reed Kelly, On a Generalization of Ramsey Theory, 2009.
Reed Kelly, Code for Computing Tuple-chromatic Ramsey Numbers and Tuple-chromatic Van der Waerden Numbers

Another part of the tcW function: A157102. The 2-color Van der Waerden numbers: A005346, W(2, k). Multi-color Van der Waerden numbers with 3-term monochrome arithmetic subsequences A135415, W(r, 3).

a(r) = tcW(r,r-1,r+1).

cp's OEIS Frontend

A100480 a(1)=1 and, for n>1, a(n) is the smallest positive integer such that the subset S(c) of {1,2,3,...,n} defined by S(c)={k|1<=k<=n; a(k)=c} does not contain a 3-term arithmetic progression for any integer c.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Formula

A157102 Tuple-chromatic Van der Waerden numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula