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
each number thus creating generation 2. Apply the same procedure over and over again to make the next generations N. The following table summarizes everything for the first nine generations:

 PRIMES Gen1 Gen2 Gen3 Gen4 Gen5 Gen6 Gen7 Gen8 Gen9 PLOT Nrs 2 3 7 13 30 56 127 237 530 prime cells 3 7 13 30 56 127 237 530 994 Odd composite 5 10 23 43 97 181 403 757 1662 7 16 30 67 125 276 520 1132 2156 11 20 44 82 179 339 729 1399 2970 13 28 52 112 214 453 879 1838 3598 17 32 68 132 274 540 1109 2199 4491 19 40 80 162 326 656 1320 2653 5335 23 48 94 194 382 780 1544 3136 6231 29 54 114 220 454 888 1816 3578 7278 31 66 126 260 506 1036 2034 4142 8178 37 72 146 286 582 1146 2326 4600 9308 41 80 160 322 640 1290 2566 5166 10290 43 88 176 354 708 1420 2840 5690 11382 2 3 3 3 4 6 5 3 Odd's

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 :
7, 23, 67, 179, 1109, 17427659, 1586875049, 7021048691
(See sequences A047844, A048448 up to A048466 at Neil's Sloane site)

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):
"I extended the search of Puzzle No. 55.

Each term of the sequence 7, 23, 67, 179, 453, 1109, ...
[This is the 'Last Odd Term' sequence] are of the form

S(n) = Sum(j = 0 to n - 1, C(n - 1, j) * P(2*j + 1))
where C(n, p) = n! / (p! * (n-p)!)
and P(1) = 2,...  P(i) is the ith prime number.

The first probable-primes of the sequence occur for n equal :
1,
2, 3, 4, 5, 7, 19, 25, 27, 53, 59, 68, 148, 176, 241, 347,
441, 444, 509, 844, 990
[found previously by Patrick, in red].
The search was extended up to 1000
"

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