Problems & Puzzles: Puzzles

Puzzle 1137 Consecutive primes & n#

Sebastián Martín Ruiz sent the following puzzle:

I have verified that for the expression Prime(n)+n#, that is prime for n=1 to 13,

where n# =  product of primes <= n. See A034386 for the definition of n# and a list of the first ones values.

First three examples are the following:

3=2+1# =2+1

5=3+2#=3+2

11=5+3#=5+3*2=5+6

All the 13 consecutive primes primes are: 3, 5, 11, 13, 41, 43, 227, 229, 233, 239, 2341, 2347, 30071

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We obtain 11 consecutive primes for the expression Prime(n+1)+n# from n=2 to 12, they are as follows

7=5+2# =5+2

13=7+3# =7+3*2=7+6

17=11+4# =11+3*2=11+6=17

All the 11 consecutive primes are: 7, 13, 17, 43, 47, 229, 233, 239, 241, 2347, 2351
 

Q) Find expressions that contain n# and generate more consecutive primes than 13.

 

During the week from July 15 -21 contributions came from Davide Rotondo, Giorgos Kalogeropoulos, JM Rebert

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Davide wrote:

a)

A341661(n) + n# produce 15 consecutive primes.
 

 
Primes p such that p^4 - 1 has fewer than 160 divisors.
 
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 59, 61, 71, 79, 101 
 

 
2+1#=2+1=3
3+2#=3+2=5
5+3#=5+6=11
7+4#=7+6=13
11+5#=11+30=41
13+6#=13+30=43
17+7#=17+210=227
19+8#=19+210=229
23+9#=23+210=233
29+10#=29+210=239
31+11#=31+2310=2341
37+12#=37+2310=2347
41+13#=41+30030= 30071
59+14#=59+30030=30089
61+15#=61+30030=30091

 

 
list of prime: 3,5,11,13,41,43,227,229,233,239,2341,2347,30071,30089,30091.

b)

 I found a better one. https://oeis.org/A296915
a296915(n) + n# produce 16 consecutive primes
 

 
 
A296915   Primes that are not squares mod 163. +40
3
  2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 59, 67, 73, 79, 89, 101, 103, 107, 109, 127, 137, 139, ....

 
2+1#=2+1=3
3+2#=3+2=5
5+3#=5+6=11
7+4#=7+6=13
11+5#=11+30=41
13+6#=13+30=43
17+7#=17+210=227
19+8#=19+210=229
23+9#=23+210=233
29+10#=29+210=239
31+11#=31+2310=2341
37+12#=37+2310=2347
59+13#=59+30030= 30089
67+14#=67+30030=30097
73+15#=73+30030=30103
79+16#=89+30030=30109

 
list: 3,5,11,13,41,43,227,229,233,239,2341,2347,30089,30097,30103,30109.

 

***

Giorgos wrote:

Prime(n+8) + n# is prime for 15 consecutive values for n=2 to 16
 
31, 37, 43, 71, 73, 257, 263, 269, 271, 2377, 2381, 30103, 30109, 30113, 30119
 

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Rebert wrote:

I found 54 consecutive primes, with a(n) = 135858383 + n#, 13 <= n <= 66.

 
n a(n)
13 135888413 is prime.
14 135888413 is prime.
15 135888413 is prime.
16 135888413 is prime.
17 136368893 is prime.
18 136368893 is prime.
19 145558073 is prime.
20 145558073 is prime.
21 145558073 is prime.
22 145558073 is prime.
23 358951253 is prime.
24 358951253 is prime.
25 358951253 is prime.
26 358951253 is prime.
27 358951253 is prime.
28 358951253 is prime.
29 6605551613 is prime.
30 6605551613 is prime.
31 200696348513 is prime.
32 200696348513 is prime.
33 200696348513 is prime.
34 200696348513 is prime.
35 200696348513 is prime.
36 200696348513 is prime.
37 7420873993193 is prime.
38 7420873993193 is prime.
39 7420873993193 is prime.
40 7420873993193 is prime.
41 304250399385593 is prime.
42 304250399385593 is prime.
43 13082761467528413 is prime.
44 13082761467528413 is prime.
45 13082761467528413 is prime.
46 13082761467528413 is prime.
47 614889782724349793 is prime.
48 614889782724349793 is prime.
49 614889782724349793 is prime.
50 614889782724349793 is prime.
51 614889782724349793 is prime.
52 614889782724349793 is prime.
53 32589158477325903113 is prime.
54 32589158477325903113 is prime.
55 32589158477325903113 is prime.
56 32589158477325903113 is prime.
57 32589158477325903113 is prime.
58 32589158477325903113 is prime.
59 1922760350154348497453 is prime.
60 1922760350154348497453 is prime.
61 117288381359407106841653 is prime.
62 117288381359407106841653 is prime.
63 117288381359407106841653 is prime.
64 117288381359407106841653 is prime.
65 117288381359407106841653 is prime.
66 117288381359407106841653 is prime.
67 7858321551080267191737473 = (113 * 69542668593630683112721) is not prime

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