Problems & Puzzles: Puzzles Puzzle 55. Primes by Generation (Patrick De Geest) Define the sequence of primes 2,
3, 5, 7, 11, 13, 17, 19, ... as generation 1. Starting from
this generation 1 add the previous and next term
of
With each generation the 'last odd term' moves down one place (see darkgreen background cells)! They form a very beautiful new sequence starting with primes but soon these become very rare: 7, 23, 67, 179, 453, 1109, 2653, 6231, 14409, 32877, 74137, 165429, 365691, 801747,... Highlighting the primes versus the composites give the following Last Odd Term sequence (primes in blue, composites in orange): 7, 23, 67, 179,453,1109, 2653, 6231, 14409, 32877, 74137, 165429, 365691, 801747, 1745331, 3776605, 8130401, 17427659, 37217597, 79224121, 168170537, 356107787, 752453861, 1586875049, 3340696135, 7021048691, 14731810645,... I was able to track down these entire Prime Last Odd Terms sequence PLOT's  up to generation 50. They occur in the following generations: 2, 3, 4, 5, 7, 19, 25, 27 (next one >50). This yield the next appealing series of eight primes
for now : a) Try to extend this sequence (find more PLOT numbers) b) Three generations (2, 3 and 4) have only 'prime' odd terms in their ranks. Exist there a fourth or even a fifth generation where this fact occurs? Solution Yves Gallot wrote (5/6/99): *** 




