Puzzle 1226 Prime + Primorials

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Puzzle 1226 Prime + Primorials

Gennady Gusev sent the following puzzle.

A. Prime + consecutive primorials.

I found that prime 41 summed with 8 consecutive primorials gives 8 primes:
1) 41 + 2# = 43 - prime
2) 41 + 3# = 47 - prime
3) 41 + 5# = 71 - prime
4) 41 + 7# = 251 - prime
5) 41 + 11# = 2351 - prime
6) 41 + 13# = 30071 - prime
7) 41 + 17# = 510551 - prime
8) 41 + 19# = 9699731 - prime

Q1. Find a longer sequence (not necessarily starting from 2#).

B. Prime + sum of consecutive primorials
I found that prime 59 summed with sum of 10 consecutive primorials gives 10 prime:
1) 59 + 2# = 59 + 2 = 61 - prime
2) 59 + 2# + 3# = 59 + 8 = 67 - prime
3) 59 + 2# + 3# + 5# = 59 + 38 = 97 - prime
4) 59 + 2# + 3# + 5# + 7# = 59 + 248 = 307 - prime
5) 59 + 2# + 3# + 5# + 7# + 11# = 59 + 2558 = 2617 - prime
6) 59 + 2# + 3# + 5# + 7# + 11# + 13# = 59 + 32588 = 32647 - prime
7) 59 + 2# + 3# + 5# + 7# + 11# + 13# + 17# = 59 + 543098 = 543157 - prime
8) 59 + 2# + 3# + 5# + 7# + 11# + 13# + 17# + 19# = 59 + 10242788 = 10242847 - prime
9) 59 + 2# + 3# + 5# + 7# + 11# + 13# + 17# + 19# + 23# = 59 + 233335658 = 233335717 - prime
10) 59 + 2# + 3# + 5# + 7# + 11# + 13# + 17# + 19# + 23# + 29# = 59 + 6703028888 = 6703028947 - prime

Q2. Find a longer sequence (not necessarily starting from 2#)

From 22 to 28, June, 2025, contributions came from Mourné Louw, Oscar Volpatti, Giorgos Kalogeropoilos, Emmanuel Vantieghem, Paul Cleary, Simon Cavegn

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Mourné wrote:

I assume we use the second definition of primorials, where n# = product of primes <= n?
If so:
Q1) Starting with the prime 235313357:
1) 235313357 + 2# = 235313359 - prime
2) 235313357 + 3# = 235313363 - prime
3) 235313357 + 5# = 235313387 - prime
4) 235313357 + 7# = 235313567 - prime
5) 235313357 + 11# = 235315667 - prime
6) 235313357 + 13# = 235343387 - prime
7) 235313357 + 17# = 235823867 - prime
8) 235313357 + 19# = 245013047 - prime
9) 235313357 + 23# = 458406227 - prime
10) 235313357 + 29# = 6705006587 - prime
11) 235313357 + 31# = 200795803487 - prime
Q2) Starting with the prime 3877889:
1) 3877889 + 2# = 3877889 + 2 = 3877891 - prime
2) 3877889 + 2# + 3# = 3877889 + 8 = 3877897 - prime
3) 3877889 + 2# + 3# + 5# = 3877889 + 38 = 3877927 - prime
4) 3877889 + 2# + 3# + 5# + 7# = 3877889 + 248 = 3878137 - prime
5) 3877889 + 2# + 3# + 5# + 7# + 11# = 3877889 + 2558 = 3880447 - prime
6) 3877889 + 2# + 3# + 5# + 7# + 11# + 13# = 3877889 + 32588 = 3910477 - prime
7) 3877889 + 2# + 3# + 5# + 7# + 11# + 13# + 17# = 3877889 + 543098 = 4420987 - prime
8) 3877889 + 2# + 3# + 5# + 7# + 11# + 13# + 17# + 19# = 3877889 + 10242788 = 14120677 - prime
9) 3877889 + 2# + 3# + 5# + 7# + 11# + 13# + 17# + 19# + 23# = 3877889 + 233335658 = 237213547 - prime
10) 3877889 + 2# + 3# + 5# + 7# + 11# + 13# + 17# + 19# + 23# + 29# = 3877889 + 6703028888 = 6706906777 - prime
11) 3877889 + 2# + 3# + 5# + 7# + 11# + 13# + 17# + 19# + 23# + 29# + 31# = 3877889 + 207263519018 = 207267396907 - prime

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Oscar wrote:

I searched for primes p which generate sequences of length n, with primorials always starting from 2#.

Q1

n p

1 3

2 5

3 11

4 17

5 101

6 21377

7 107

8 41

9 37872221

10 86351

11 235313357

12 729457511

13 99445156397

14 818113387907

15 7986903815771

Q2

1 17

2 11

3 5

4 3

5 101

6 19469

7 38669

8 191459

9 191

10 59

11 3877889

12 494272241

13 360772331

14 6004094833991

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Giorgos wrote:
Q1.
1) 99445156397 + 2# = 99445156399 - prime
2) 99445156397 + 3# = 99445156403 - prime
3) 99445156397 + 5# = 99445156427 - prime
4) 99445156397 + 7# = 99445156607 - prime
5) 99445156397 + 11# = 99445158707 - prime
6) 99445156397 + 13# = 99445186427 - prime
7) 99445156397 + 17# = 99445666907 - prime
8) 99445156397 + 19# = 99454856087 - prime
9) 99445156397 + 23# = 99668249267 - prime
10) 99445156397 + 29# = 105914849627 - prime
11) 99445156397 + 31# = 300005646527 - prime
12) 99445156397 + 37# = 7520183291207 - prime
13) 99445156397 + 41# = 304349708683607 - prime

Q2.1) 360772331 + 2# = 360772333 - prime
2) 360772331 + 2# + 3# = 360772339 - prime
3) 360772331 + 2# + 3# + 5# = 360772369 - prime
4) 360772331 + 2# + 3# + 5# + 7# = 360772579 - prime
5) 360772331 + 2# + 3# + 5# + 7# + 11# = 360774889 - prime
6) 360772331 + 2# + 3# + 5# + 7# + 11# + 13# = 360804919 - prime
7) 360772331 + 2# + 3# + 5# + 7# + 11# + 13# + 17# = 361315429 - prime
8) 360772331 + 2# + 3# + 5# + 7# + 11# + 13# + 17# + 19# = 371015119 - prime
9) 360772331 + 2# + 3# + 5# + 7# + 11# + 13# + 17# + 19# + 23# = 594107989 - prime
10) 360772331 + 2# + 3# + 5# + 7# + 11# + 13# + 17# + 19# + 23# + 29# = 7063801219 - prime
11) 360772331 + 2# + 3# + 5# + 7# + 11# + 13# + 17# + 19# + 23# + 29# + 31# = 207624291349 - prime
12) 360772331 + 2# + 3# + 5# + 7# + 11# + 13# + 17# + 19# + 23# + 29# + 31# + 37# = 7628362426159 - prime
13) 360772331 + 2# + 3# + 5# + 7# + 11# + 13# + 17# + 19# + 23# + 29# + 31# + 37# + 41# = 311878625953369 - prime
Later he added...

I realised now that this is already in OEIS
If we start with 2# then both questions have answers in OEIS:
Q1 is A115786 with record 29065965967667

Q2 is A257466 with record 41320119600341

For your convenience I copy here below part of these sequences, cited by Giorgos:

A115786

Smallest prime number p such that p + 2#, p + 3#, ..., p + prime(n)# are all prime, where x# = A034386(x) is the primorial.

2

3, 5, 11, 17, 41, 41, 41, 41, 86351, 86351, 235313357, 729457511, 99445156397, 818113387907, 7986903815771, 29065965967667

Authors:Rick L. Sheperd, Don Reble & J. K. Andersen

A257466

Smallest prime number p such that p + pps(1), p + pps(2), ..., p + pps(n) are all prime but p + pps(n+1) is not, where pps(n) is the partial primorial sum (A060389(n)).

2

2, 17, 11, 5, 3, 101, 19469, 38669, 191459, 191, 59, 3877889, 494272241, 360772331, 6004094833991, 41320119600341

Author: Fred Schneider

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Emmanuel wrote:

First group starting with 2#.

The following ten sums are all prime :

1) 86351 + 2#
2) 86351 + 3#
3) 86351 + 5#
4) 86351 + 7#
5) 86351 + 11#
6) 86351 + 13#
7) 86351 + 17#
8) 86351 + 19#
9) 86351 + 23#
10) 86351 + 29#

First group, not starting with 2#.

The following 13 sums are all prime :
1) 135858383 + 13#
2) 135858383 + 17#
3) 135858383 + 19#
4) 135858383 + 23#
5) 135858383 + 29#
6) 135858383 + 31#
7) 135858383 + 37#
8) 135858383 + 41#
9) 135858383 + 43#
10) 135858383 + 47#
11) 135858383 + 53#
12) 135858383 + 59#
13) 135858383 + 61#

Second group starting with 2#.

The following 11 sums are all prime :
1) 3877889 + 2#
2) 3877889 + 2# + 3#
3) 3877889 + 2# + 3# + 5#
4) 3877889 + 2# + 3# + 5# + 7#
5) 3877889 + 2# + 3# + 5# + 7# + 11#
6) 3877889 + 2# + 3# + 5# + 7# + 11# + 13#
7) 3877889 + 2# + 3# + 5# + 7# + 11# + 13# + 17#
8) 3877889 + 2# + 3# + 5# + 7# + 11# + 13# + 17# + 19#
9) 3877889 + 2# + 3# + 5# + 7# + 11# + 13# + 17# + 19# + 23#
10) 3877889 + 2# + 3# + 5# + 7# + 11# + 13# + 17# + 19# + 23# + 29#
11) 3877889 + 2# + 3# + 5# + 7# + 11# + 13# + 17# + 19# + 23# + 29# + 31#

Second group not starting with 2#

This can mean two things :

* the first sum is of the form p + 2# + ... +q#
* the first sum is of the form p + q#
In the first case I found that the following 12 sums are all prime:
1) 71292803 + 2# + 3# + 5#
2) 71292803 + 2# + 3# + 5# + 7#
3) 71292803 + 2# + 3# + 5# + 7# + 11#
4) 71292803 + 2# + 3# + 5# + 7# + 11# + 13#
5) 71292803 + 2# + 3# + 5# + 7# + 11# + 13# + 17#
6) 71292803 + 2# + 3# + 5# + 7# + 11# + 13# + 17# + 19#
7) 71292803 + 2# + 3# + 5# + 7# + 11# + 13# + 17# + 19# + 23#
8) 71292803 + 2# + 3# + 5# + 7# + 11# + 13# + 17# + 19# + 23# + 29#
9) 71292803 + 2# + 3# + 5# + 7# + 11# + 13# + 17# + 19# + 23# + 29# + 31#
10) 71292803 + 2# + 3# + 5# + 7# + 11# + 13# + 17# + 19# + 23# + 29# + 31# + 37#
11) 71292803 + 2# + 3# + 5# + 7# + 11# + 13# + 17# + 19# + 23# + 29# + 31# + 37# + 41#
12) 71292803 + 2# + 3# + 5# + 7# + 11# + 13# + 17# + 19# + 23# + 29# + 31# + 37# + 41# + 43#
In the second case I found that the following 14 sums are prime :
1) 65987791 + 3#
2) 65987791 + 3# + 5#
3) 65987791 + 3# + 5# + 7#
4) 65987791 + 3# + 5# + 7# + 11#
5) 65987791 + 3# + 5# + 7# + 11# + 13#
6) 65987791 + 3# + 5# + 7# + 11# + 13# + 17#
7) 65987791 + 3# + 5# + 7# + 11# + 13# + 17# + 19#
8) 65987791 + 3# + 5# + 7# + 11# + 13# + 17# + 19# + 23#
9) 65987791 + 3# + 5# + 7# + 11# + 13# + 17# + 19# + 23# + 29#
10) 65987791 + 3# + 5# + 7# + 11# + 13# + 17# + 19# + 23# + 29# + 31#
11) 65987791 + 3# + 5# + 7# + 11# + 13# + 17# + 19# + 23# + 29# + 31# + 37#
12) 65987791 + 3# + 5# + 7# + 11# + 13# + 17# + 19# + 23# + 29# + 31# + 37# + 41#
13) 65987791 + 3# + 5# + 7# + 11# + 13# + 17# + 19# + 23# + 29# + 31# + 37# + 41# + 43#
14) 65987791 + 3# + 5# + 7# + 11# + 13# + 17# + 19# + 23# + 29# + 31# + 37# + 41# + 43# + 47#

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Paul wrote:

Only had time to attempt q1 this week, managed to get a sequence 13 long.

RunLength->13,

InitialPrime->135858383,

StartingPrimorialIndex->#6,

PrimeSequence->

135858383,

135888413,

136368893,

145558073,

358951253

6605551613,

200696348513,

7420873993193,

304250399385593,

13082761467528413,

614889782724349793,

32589158477325903113,

1922760350154348497453,

117288381359407106841653.

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Simon wrote:

Q1
22635147439 + 31# length:15
181021033469 + 5# length:15

Q2
65987789 + 2# length:15

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