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.  

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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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Giovanni Resta wrote on Jan 28, 2020:

for what concerns Puzzle 782 question Q2,
J. K. Andersen proposed to find primes of the form p + k! for k=2,...,n,

and he found terms up to n=13 p=35548069540727.
This is sequence A256301 in OEIS https://oeis.org/A256301

Recently Khalid Sabry has found n=14 p=1490524895687 and
I have extended it to n=15, p = 13101487760627087.

That is, 13101487760627087 and 13101487760627087 + k! is prime for all k
from 2 to 15.

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Davide Rotondo wrote on July 25, 2022:

((40*(n) - 5 + 3*(mod(n,5)) + 3*(mod((n+1),5)) - 2*(mod((n+2),5)) + 3*(mod((n+3),5)) - 7*(mod((n+4),5)))/25)^2 - 79*(40*(n)- 5 + 3*(mod(n,5)) + 3*(mod((n+1),5)) - 2*(mod((n+2),5)) + 3*(mod((n+3),5)) - 7*(mod((n+4),5)))/25 + 1601

1447, 1373, 1231, 1163, 1097, 911, 853, 743, 691, 641, 503, 461, 383, 347, 313, 223, 197, 151, 131, 113, 71, 61, 47, 43, 41, 47, 53, 71, 83, 97, 151, 173, 223, 251, 281, 383, 421, 503, 547, 593, 743, 797, 911, 971, 1033, 1231, 1301, 1447, 1523, 1601, 1847, 1933, 2111, 2203, 2297, 2591, 2693, 2903, 3011, 3121, 3463, 3581, 3823, 3947, 4073, 4463, 4597, 4871, 5011, 5153, 5591, 5741, 6047, 6203, 6361

produces 75 consecutive (succesive) primes (65 distinct and 10 repeated in bold letters) for n from 1 to 75.

I (CR) believe that this the first real improvement in several years about the issue of Q2, namely "Non-Polynomial succesive-primes generator functions", at least not classical polynomials due to the use of modular operators.

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Dmitry Kamenetsky wrote on July 30, 2022:

Perhaps you would be interested in my earlier prime generating function:

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Sebastian Martin Ruiz wrote on July 30, 2022:

This is a non-polynomial function that ONLY generate (infinite?) primes, all distinct except the prime "2" that is repeated many times:
 
GCD[2n + 1, n! + 1] + Floor[1/GCD[2n + 1, n! + 1]]

Here are all the primes>2, for n=1 to 300

F[n_]:=GCD[2n+1,n!+1]+Floor[1/GCD[2n+1,n!+1]]

Do[If[F[n]>2,Print[n," ",F[n]]],{n,1,300}]

n, prime

3 7

5 11

9 19

21 43

23 47

33 67

39 79

51 103

63 127

65 131

81 163

89 179

95 191

99 199

113 227

131 263

173 347

183 367

191 383

209 419

215 431

221 443

239 479

245 491

251 503

261 523

281 563

285 571

299 599

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On August 4, 2022 Simon Cavegn wrote:

Using integer division (rounded down), f(n) is prime for n=0..102:
f(n)=41619623+n*103123020+n/13*(-1237476240)+n/26*(-65763840)+n/39*(-37359180)+n/65*2663271128

This interesting function produces 103 primes, in the range n=0 to 102. These are 4 primes unique, 15 primes that appears twice, and 23 primes that appears three times (4+15*2+23*3 = 4+30+69 = 103) 

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See more functions alike in Puzzle 1097 and 1099

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