Problems & Puzzles: Puzzles Puzzle 66. The SOPF sequences Patrick De Geest defined the sequence C_{1}, C_{2}, C_{3}, …C_{L}, P, as a succession of L composite numbers and an ending prime, where every number is the sum of the previous one plus the sum of all his prime factors (repetitions included). Just in order to be precise, let C_{i} =p_{1}^{a1}.p_{2}^{a2}… , then C_{i+1} = C_{i }+ (sum of) (a_{j} . p_{j}). Example (and the least of this kind of sequences): 4,
8, 14, 23 Questions (1, 3 & 5 by C. Rivera; 2 & 4 by Patrick De Geest): 1. Find three consecutive composites C_{1}, C_{1}+1 and C_{1}+2 such that all they have the same ending prime in their respective sequence, or prove that such triplet cannot exist. 2. Find the first occurrence of a chain of 20=>Z=>2 consecutive numbers with the same parity (this chain may be a part of a complete sequence)
3. Find the least complete sequences formed with Z odd members, for Z = 6, 7, 8, 9 & 10.
5. Defining the "Rate of Growth of C_{1}", as RG(C_{1}) = (P/C_{1})/L, I found the following conspicuous C_{1}: RG(137222) = (35200439/137222)/99 = 2.59113 Last weekend I asked to certain friends of this site if they could improve this rate. The 9/9/99 Patrick De Geest responded: RG(13161303) = (31760920439/13161303)/216 = 11.1722 Can you improve the Patrick's result? Jud McCranie four triplets of consecutive composite numberssolution for the part 1 of this puzzle: 559561; 19461948; 31553157; 1722617228. Jud McCranie found some solutions to part 2. of this puzzle: Z = 9: Starting at 1099 Jud McCranie & Enoch Haga independently found solutions to part 3. of this puzzle. See their contributions in the table above. *** Anton Vrba wrote on June 2011:
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