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February 28th, 2016, 03:29 AM   #1
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Prime numbers

Hi,

Here is a sequence of prime numbers:
3,5,11,17,47,257,510767,....
The algorithm used to build such sequence is :
We define primorial p noted #p=2*3*5*7*....*p
Start from U(0)=3 the first odd prime
Compute a(1)=int(sqrt(U(0)))+1=2
U(1)=U(0)+a(1) where a(1) is equal to primorial #2 = 2
U(1)=5
Compute a(2)=int(sqrt(U(1))+1=3
U(2)=U(1)+a(2)=5+#3=5+(2*3)=11
Compute a(3)=int(sqrt(11))+1=4 (4 is not prime so we use #3=2*3
U(3)=11+(2*3)=17
and so on

Are all the numbers of the sequence prime numbers?
The sequence is growing faster it is not easy to know.
Thank you for any help.

Last edited by skipjack; February 28th, 2016 at 05:01 AM.
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February 28th, 2016, 04:58 AM   #2
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Can you post some further numbers in the sequence?
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February 28th, 2016, 06:35 AM   #3
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The next number is 300 digits size.
So the sequence is exploding. I can not even estimate the digit size of the further numbers.
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February 28th, 2016, 07:11 AM   #4
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It is not so useful because some prime numbers are missing in the sequence. Maybe tweaking the algorithm a little bit accordingly can give all prime numbers.
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February 28th, 2016, 08:25 AM   #5
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I'm sorry, but I can't confirm that U(7) is a prime number:

Code:
(18:18) gp > U6=510767
%40 = 510767
(18:18) gp > A7=sqrtint(U6)+1
%41 = 715
(18:18) gp > V7=primes(primepi(A7))
%42 = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149,151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709]
(18:19) gp > U7=U6+factorback(V7)
%43 = 13802651106711802536344050306133362992649963656229914863058580142142610482430817949922104531639351381921564573865712490763569228376295661814770390189505137031692781919527713285374540164408571278055683171593020170233128086464775974520546386806644011091046992108509661969860784773011026549130272637
(18:20) gp > isprime(U7)
%44 = 0
According to my calculations U(7) is composite.
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February 28th, 2016, 09:26 AM   #6
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U(7) is composite but what about the other computable terms.
@Prakhar It is not easy to find an infinite sequence of only prime numbers. I hope that my sequence up to U(10000) contains 90% of prime numbers I will be then very happy
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