Problems & Puzzles: Puzzles

Puzzle 782. Prime-Generating non-polynomials

Jaime Ayala suggests to create a catalog of Prime-Generating record non-polynomials, catalog similar to these Prime-Generating record polynomials summarized in the table of the following well known Wolfram Mathworld page.

Ayala contributes with one specific non-polynomial form: P+p#n, where P is a prime number and p#n is the primorial function 2*3*...*pn.

Ayala noticed that for P=41 this form generates 8 primes in a row for the first 8 primorials, that is to say, 41+2#, 41+3#, 41+5#, ..., 41+19# are the following prime numbers: 43, 47, 71, 251, 2351, 30071, 510551, 969931.

Carlos Rivera found that the non-polynomial form studied by Ayala provides 12 primes in a row for P= 729457511. That is to say 729457511+2#, 729457511+3#+...+729457511+37# are the following prime numbers: 729457513, 729457517, 729457541, 729457721, 729459821, 729487541, 729968021, 739157201, 952550381, 7199150741, 201289947641, 7421467592321. No more than 12 primes in a row are generated for P<2^32.

Q1. Send your prime P if it generates more than 12 primes in a row for the form P+p#n.

Q2. Perhaps you'll like to contribute with another type of Prime-Generating record non-polynomial.


Contributions came from J. K. Andersen, Maximilian Hasler, Jan van Delden and Emmanuel Vantiegem.

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Andersen wrote:

https://oeis.org/A115786 is "Smallest prime number p such that p + 2#, p + 3#,
..., p + prime(n)# are all prime".
3, 5, 11, 17, 41, 41, 41, 41, 86351, 86351, 235313357, 729457511, 99445156397,
818113387907, 7986903815771, 29065965967667

729457511 is term 12 from Don Reble. I found term 13 to 16 in 2006 when Puzzle 350 was published about the corresponding sequence A115785 for p - prime(n)#.
 

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Maximilian contributed two more polynomial examples more appropriate for the mentioned Table in the Wolfram's article.

a) 6*n^2+17, is prime for n=0 to 16
b) 3*n^2+3*n+23, is prime for n=0 to 21

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Jan wrote:

Q1: Smallest solutions:
 
n=13 p=99445156397
n=14 p=818113387907
 
I searched until 7204197000000.
 
Q2:
 
P(k)=p+(k+1)! prime for k=1..n
 
Smallest solutions:
n=12 p=79017245897
n=13 p=35548069540727
 
I searched until 72072000000000.

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Emmanuel wrote:

Here is a prime  p  such that  p + pn#  is prime for  n = 1, 2, 3, ..., 13 : p = 99445156397.
 
These are those primes :
99445156399, 99445156403, 99445156427, 99445156607, 99445158707, 99445186427, 99445666907, 99454856087, 99668249267, 105914849627, 300005646527, 7520183291207, 304349708683607
 
If I made no mistakes, there is no smaller  p  with that property but I believe that there are many bigger ones. Only : they are too big for my programming capabilities ...

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Fred Schneider wrote on May 25, 2015:

I have two submissions for 2)

 a) Partial primorial sum (
https://oeis.org/draft/A257466)

 
If x is prime. search for prime sequences of the form:
 x + 2, x + 2 + 2*3, x + 2 + 2*3 + 2*3*5, etc
In other words, the difference between successive terms is the primorial

 
I found the first 14 terms in the sequence:

 
The 14th term, 6004094833991, yields these 14 primes
 

[6004094833993,6004094833999,6004094834029,6004094834239,6004094836549,6004094866579,
6004095377089,6004105076779,6004328169649,6010797862879,6211358353009,13632096487819,
317882360015029,13400643691685059

The 15th value in that sequence is composite:
628290426280176469 = 45953 * 13672457212373 


b) Primorial Square (
https://oeis.org/draft/A257467)

If x is prime. search for prime sequences of the form:
 x + 2^2, x + (2*3)^2, x + (2*3*5)^2

I found the first 12 terms in the sequence:

The 12th term in the sequence: 37520993053 yields these 12 primes:

37520993057,37520993089,37520993953,37521037153,37526329153,38422793953,298141453153,
94121507089153,49770466165829953,41856930527828825953,40224510201223348409953,
55067354465423435254729153

The 13th value in that sequence is composite:

92568222856376731627931377153 = 227*73613*95143*58224358260055921


Perhaps someone would like to try extending them.  

***

Later, on May 15 2015, he added:

c) Partial sum of square of primorial:  https://oeis.org/A258035
I found 13 terms.  The 13th is: 4428230508349

 

[4428230508349, 4428230508353,4428230508389,4428230509289,4428230553389,4428235889489,
4429137690389,4689758150489,98773744246589,49869202389083489,41906799692696916389,
40266417000878524333289,55107620882424276258069389,92623330477259155866668453489]

The 14th term in that sequence is composite:
 

171251267391917835866535468654389 = 71 * 357439202389 * 6747971866139479831

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 Arkadiusz Wesolowski wrote on Jan 31, 2016:

Here is my best solution for this puzzle (Q2).
 
The value of ((-1)^n + 2*n - 38)*(2*n - 38) + 41 is prime for 0 <= n <= 59. So we get 60 distinct primes (http://oeis.org/A226097).

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