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A157102 Tuple-chromatic Van der Waerden numbers. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

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)

LINKS

Table of n, a(n) for n=2..61.

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

FORMULA

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

EXAMPLE

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.

MATHEMATICA

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

PROG

(Other) A C++ program is available from the links. It is not the best program, but it is relatively fast. To get the terms of the above sequence, you have to compile the program and choose parameters such as: find_vdw 10000 5 4 5 for tcW(5, 4, 5) and find_vdw 10000 6 5 6 for tcW(6, 5, 6).

CROSSREFS

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).

Sequence in context: A160994 A113833 A212286 * A226512 A270307 A261480

Adjacent sequences:  A157099 A157100 A157101 * A157103 A157104 A157105

KEYWORD

nonn

AUTHOR

Reed Kelly, Feb 22 2009, Feb 25 2009

STATUS

approved

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Last modified September 21 19:59 EDT 2019. Contains 327282 sequences. (Running on oeis4.)