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 On-Line integer sequences

“Are any sequences in Sloane's Integer Sequences unexpectedly rich in primes??”

b) 'The Unknown Sloane'

If one searches Sloane's Integer Sequence Online Encyclopedia
for the 'unkn' keyword around 8 entries will pop up. The most attractive ones are in my opinion the following two:A036235 and A020993

A036235:{5,22,121,496,...}
A020993:{100,85,80,76,70,65,61,60,56,55,52,...}

Carlos, this is my puzzle question (or Sloane's!).

Find valid 'rules' that describes in a unique way
the terms in sequences A036235 and A020993.

If it turns out one can find rules related with primes, well then,
wouldn't that be nice for you PP&P pages !. Who can find the most original, the most beautiful description for these terms ?

Jim R. Howell (13/02/99) has found a solution to the 'unknown' Sloane's sequence A036235. This is his e-mail:

"One way to define the Sloane sequence { 5, 22, 121, 496, ... } is that the n-th term is: (97*n^3 - 168*n^2 + 122*n + 15) / 3, (where "5" is the 0-th term). With this definition, the first several terms are:

{ 5, 22, 121, 496, 1341, 2850, 5217, 8636, 13301, ...}

Since this is a PRIMES puzzle page, the first few primes in this sequence are:

5 (n=0), 118757 (n=16), 171161 (n=18), 531497 (n=26), 667021 (n=28), 17454361 (n=82), and 21604181 (n=88).

I also noticed the following interesting thing about this sequence: There are 5 primes between 5 and 22 (7, 11, 13, 17, and 19), and there are 22 primes between 22 and 121. However, there are only 64 primes between 121 and 496. The "496" would need to be replaced by 880 (or 878 or 879) in order to have 121 primes between 121 and the next number in the sequence."

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
S[1] = 2 * Prime[5]
S[2] = [Prime[5]]^2
S[3] = 2^4 * Prime[Prime[5]]
S[4] = [Prime[Prime[5]]]^(2^4)
....
S[2n-1] = (2^(n^2)) * Prime^n[S[0]]
S[2n] = Prime^n[S[0]]^(2^(n^2))

Where Prime^n[] means apply the prime function repeatedly.

S[5] = 127 * 2^9
S[6] = 127^(2^9)
S[7] = 709 * 2^16
S[8] = 709^(2^16)
S[9] = 5381 * 2^25
S[10] = 5381^(2^25)

…as you can see it grows quite rapidly.

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On Feb, 17, 2022 Giorgos Kalogeropoulos wrote:

Recently I published a sequence in oeis: https://oeis.org/A350545

Please read my conjecture in the comments.

I believe that this sequence qualifies as an answer for the first question of puzzle 38 where you ask:

“Are any sequences in Sloane's Integer Sequences unexpectedly rich in primes??”

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