We need Tao, Green and Maynard

 

Tao, Green and Maynard

Inspired by the beautiful graphs computed by Giorgos Kalogeropoulos here, we had the idea to replace the Champernowne decimal expansion (A033370) by the successive primes (A000040). Big mistake.

Here are the first prime numbers:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67,…

We want to produce them one by one with the successive sums a(n) + a(n+1) of a new sequence T – and, just for the fun, we want all terms of T to be distinct nonprimes.

Let us try a first approach with a(1) = 1.

Well, no sum 1 + a(2) = 2 is possible (we cannot use 1 again). We have to use a negative a(1). We try a(1) = -1.

But this does not work, as -1 + a(2) = 2 would produce a(2) = 3, a prime.

Ok, we quickly see that a(1) = 0 does not work either, nor a(1) = -4 [as -4 would produce a(2) = 6, ok, but a(2) = 6 would produce a(3) = -3, a negative prime]. We are stuck.

Digging into the negative values, we then try a(1) = -6. This leads to a(2) = 8, ok, but a(2) = 8 leads to a(3) = -5… forbidden!

We try then a(1) = -8. Which leads to a(2) = 10, yes, but a(2) = 10 leads to a(3) = -7… a prime again.

What about a(1) = -9?

a(1) = -9 produces a(2) = 11, a prime.

a(1) = -10 produces a(2) = 12, which is a composite number, good, and a(2) = 12 would lead to a(3) = -9, a composite again! We see that a(3) = -9 leads to -9 + a(4) = 5, leading itself to a(4) = 14, another composite! We are making some progresses here – we have so far:

T = -10, 12, -9, 14, …

But the next term of T will be -7…

We have (manually) explored this sequence for a(1) = 0 to a(1) = -100… without success: all chains stop, the longest one having only 9 terms!

We have the intuition that no such infinite sequence T is possible – and will call soon Tao, Green and Maynard to be sure.

Giorgos Kalogeropoulos has computed the lexicographically earliest sequence of distinct nonprimes such that the successive sums a(n) + a(n+1) are the first 100 primes (this will be soon in the OEIS, here):

T = -369123, 369125, -369122, 369127, -369120, 369131, -369118, 369135, -369116, 369139, -369110, 369141, -369104, 369145, -369102, 369149, -369096, 369155, -369094, 369161, -369090, 369163, -369084, 369167, -369078, 369175, -369074, 369177, -369070, 369179, -369066, 369193, -369062, 369199, -369060, 369209, -369058, 369215, -369052, 369219, -369046, 369225, -369044, 369235, -369042, 369239, -369040, 369251, -369028, 369255, -369026, 369259, -369020, 369261, -369010, 369267, -369004, 369273, -369002, 369279, -368998, 369281, -368988, 369295, -368984, 369297, -368980, 369311, -368974, 369321, -368972, 369325, -368966, 369333, -368960, 369339, -368956, 369345, -368948, 369349, -368940, 369359, -368938, 369369, -368936, 369375, -368932, 369381, -368924, 369385, -368922, 369389, -368910, 369397, -368906, 369405, -368902, 369411, -368890, 369413, -368872, ...

Meanwhile we had the idea to turn this failure into another sequence: a(n) would be the smallest integer leading to a (halting) chain of n terms…

____________________

... and Giorgos was quick to compute the new U sequence:

U = -2, -1, -4, -25, -10, -119, -28, -91, -96, -121, -122, -205, -302, -117, -124, -287, -190, -695, -222, -663, -556, -1333, -1358, -527, -1274, -1375, -1140, -2033, -1086, -3933, -3178, -5203, -1008, -763, -5296, -2453, -3294, -1331, -1398, -7693, -2514, -9541, -3998, -9979, -7094, -3773, -4432, -11363, -5838, -4291, -4886, -14023, -5970, -6391, -8536, -17233, -23490, -9313, -33978, -78073, -26548, -38019, -35708, -16239, -50738, -44323, -35746, -60839, -69648, -29635, -69708... 

... precisely, Giorgos, bravo and many thanks – I will submit this to the OEIS soon!