Problems & Puzzles: Puzzles

 Puzzle 634. Primes in Collatz trajectory This last week I have been playing in private, a game induced by some communications from my friend G. L. Honaker, Jr.The game is to get the champions of integers n such that its Collatz trajectory (until it lands on the prime "2") produces more primes than all integers m less than n. As a matter of fact, this game has been played since 2002 by other players (Joseph Le Pe and Donovan Johonson) and their results, at the beginning of this week were the following ones (See A078373): 2, 3, 7, 19, 27, 97, 171, 231, 487, 763, 1071, 4011, 6171, 10971, 17647, 47059, 99151, 117511, 202471, 260847, 481959, 963919, 1564063, 1805311, 1993215, 6991599, 8400511, 11200681, 36791535, 46564287 This week I produced three more terms: 103359483, 206718967 & 359502063 (which has been sent to OEIS) and Jud McCranie (who also was playing this game with G.L.) produced the following terms: 1376572231, 1476391215 & 2952782431 (have these been sent to OEIS?) BTW, the quantity of primes produced by 2952782431 in the whole Collatz trajectory is 87. Q. Can you produce more terms for this sequence?

Contributions came from Jud McCranie, Seiji Tomita

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Jud wrote:

I have added these record-breaking terms, and that is probably as far as I'm going.

1376572231 (82)
1476391215 (84)
2378888295 (85)
2585880735 (86)
2952782431 (87)
6423850623 (90)
12847701247 (92)

And 26404836711 has 93 primes.  It is probably the smallest one with 93 primes in the trajectory, but I can't be sure because some smaller ones got terms too large to test... my current program can only prove numbers prime if they are less than 2^62. A few of the terms got larger than that.

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Seiji wrote:

6423850623 produces 90 primes.

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