This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A157102 #12 Dec 12 2024 10:06:45
%S A157102 3,7,7,21,11,43,15,25,19,111,23,157,27,43,31,273,35,343,39,61,43,507,
%T A157102 47,121,51,79,55,813,59,931,63,97,67,171,71,1333,75,115,79,1641,83,
%U A157102 1807,87,133,91,2163,95,337,99,151,103,2757,107,271,111,169,115,3423,119,3661
%N A157102 Tuple-chromatic Van der Waerden numbers.
%C A157102 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.
%C A157102 a(r) = tcW(r,r-1,r).
%H A157102 Reed Kelly, <a href="http://www.keldesign.com/math/TCRamsey/Tuple-Chromatic-Ramsey.pdf">On a Generalization of Ramsey Theory</a>, 2009.
%H A157102 Reed Kelly, <a href="http://www.keldesign.com/math/TCRamsey/Code/index.html">Code for Computing Tuple-chromatic Ramsey Numbers and Tuple-chromatic Van der Waerden Numbers</a>
%F A157102 a(n) = (n-1)*(smallest prime factor of n) + 1.
%e A157102 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).
%e A157102 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.
%e A157102 a(2) = (2-1)(2) + 1 = 3 a(15) = (15-1)(3) + 1 = 43.
%t A157102 Table[(x - 1) * (FactorInteger[x])[[1]][[1]] + 1, {x, 2, 100}]
%o A157102 (C++) See Kelly link. 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).
%Y 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).
%K A157102 nonn
%O A157102 2,1
%A A157102 _Reed Kelly_, Feb 22 2009, Feb 25 2009