Problems & Puzzles: Puzzles

Puzzle 399. Some more terms Farideh asks: Maybe some of your puzzlers can find some more terms of the following three nice sequences:  A090872, A090874 & A090875.

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Contributions came from J. K. Andersen & Giovanni Resta:

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Andersen wrote (6/5/07):

A090872(6) is 2072005925466, since these are primes:
2072005925466^1+1
2072005925466^2+1
2072005925466^4+1
2072005925466^8+1
2072005925466^16+1
2072005925466^32+1
2072005925466^64+1

The solution can also be added to puzzle 137. Yves Gallot's estimate of 10^14 in puzzle 137 was too high.

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Giovanni wrote (11/5/07)

the next term of A090872  = a(n), is the smallest number m greater than 1 such that m^2^k+1 for k=0,1,...,n are primes is 2072005925466. I stopped my search there.

I've searched also for A090875  = a(n) is the smallest number m such that m^(2^k) + 2^(2^k) is prime for k=0,1,...,n. up to 2,365,100,000,000 but I did not find the next term.
 

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