Problems & Puzzles: Puzzles Puzzle 38. “Sloane’s sequences” This time is the turn for two puzzles suggested to me and related with the very interesting site Sloane`s OnLine integer sequences The first one is an idea of Ed Pegg, Jr. (7/1/99) www.mathpuzzle.com a) Unexpected rich in primes Sloane’s sequences “Are any sequences in
Sloane's Integer Sequences unexpectedly rich in
primes??” Patrick De Geest (12/1/99) proposed the second one b)
'The Unknown Sloane' The sequences are: A036235:{5,22,121,496,...} Solution Jim R. Howell (13/02/99) has found a solution to the 'unknown' Sloane's sequence A036235. This is his email: "One way to define the Sloane sequence { 5, 22,
121, 496, ... } is that the nth term is:
(97*n^3  168*n^2 + 122*n + 15) / 3, (where "5"
is the 0th term). With this definition, the first
several terms are: *** Jud McCranie (14/2/99) sends the corresponding polynomial solution to the sequence A020993 " (53x^10 + 3125x^9  80190x^8 + 1175490x^7  10857189x^6 + 65681805x^5  261747160x^4 + 671977660x^3  1049105808x^2 + 878094720x)/604800  388 for n=1 to 11 gives the first 11 terms of sequence A020993. Adding that "However, this is almost certainly not the solution N. J. A. Sloane is seeking". As a matter of fact, Neil Sloane  and also Jim Howell  recongize that the polynomial solution is not the intended solution to this kind of puzzles.So better solutions should come in the near future to improve the current ones. Who knows?... *** Unexpectedly five years later came a very smart and interesting solution for one of the puzzling sequences asked by Patrick (A036235). The solver was Rickey Bowers Jr., who wrote (Jan 2005): I have stumbled upon a very prime centric solution to the first series: A036235:{5,22,121,496,...} The fact of Prime[5] = 11, and Prime[11] = 31 is too strong to ignore.
S[0] = 5 Where Prime^n[] means apply the prime function repeatedly.
S[5] = 127 * 2^9 …as you can see it grows quite rapidly. ***





