Problems & Puzzles: Puzzles

Puzzle 1147 Successive prime using the functions.... On Oct. 10, 2023, Sebastián Martín Ruiz wrote: Let be the function F[n]=Prime[n]+PrimePi[n+1]! where PrimePi[n] is the prime counting function. F[n] is prime for n=1 to 13 Prime[1] + PrimePi[2]! = 2 + 1 != 2 + 1 = 3 Prime Prime[2] + PrimePi[3]! = 3 + 2 != 3 + 2 = 5 Prime Prime[3] + PrimePi[4]! = 5 + 2 != 5 + 2 = 7 Prime Prime[4] + PrimePi[5]! = 7 + 3 != 7 + 6 = 13 Prime Prime[5] + PrimePi[6]! = 11 + 3 != 11 + 6 = 17 Prime Prime[6] + PrimePi[7]! = 13 + 4 != 13 + 24 = 37 Prime Prime[7] + PrimePi[8]! = 17 + 4 != 17 + 24 = 41 Prime Prime[8] + PrimePi[9]! = 19 + 4 != 19 + 24 = 43 Prime Prime[9] + PrimePi[10]! = 23 + 4 != 23 + 24 = 47 Prime Prime[10] + PrimePi[11]! = 29 + 5 != 29 + 120 = 149 Prime Prime[11] + PrimePi[12]! = 31 + 5 != 31 + 120 = 151 Prime Prime[12] + PrimePi[13]! = 37 + 6 != 37 + 720 = 757 Prime Prime[13] + PrimePi[14]! = 41 + 6 != 41 + 720 = 761 Prime Prime[14] + PrimePi[15]! = 43 + 6 != 43 + 720 = 763 Composite. 763=7·109 Q) Find Series with expressions with Prime[n] and PrimePi[n] that generate more than 13 successive prime numbers.

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During the week from Oct 21 to Oct 27 contributions came from JM Rebert and Alain Rochelli, Oscar Volpatti

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JM Rebert wrote:

The function:

F(n) = Prime(n + 496) - 30 * PrimePi(n + 648) 

is prime for n = 1 to 38.

 

n    F(n)

 1   7 = 3547 - 30 * 118

 2  17 = 3557 - 30 * 118

 3  19 = 3559 - 30 * 118

 4  31 = 3571 - 30 * 118

 5  11 = 3581 - 30 * 119

 6  13 = 3583 - 30 * 119

 7  23 = 3593 - 30 * 119

 8  37 = 3607 - 30 * 119

 9  43 = 3613 - 30 * 119

10  47 = 3617 - 30 * 119

11  23 = 3623 - 30 * 120

12  31 = 3631 - 30 * 120

13   7 = 3637 - 30 * 121

14  13 = 3643 - 30 * 121

15  29 = 3659 - 30 * 121

16  41 = 3671 - 30 * 121

17  43 = 3673 - 30 * 121

18  47 = 3677 - 30 * 121

19  61 = 3691 - 30 * 121

20  67 = 3697 - 30 * 121

21  71 = 3701 - 30 * 121

22  79 = 3709 - 30 * 121

23  89 = 3719 - 30 * 121

24  97 = 3727 - 30 * 121

25  73 = 3733 - 30 * 122

26  79 = 3739 - 30 * 122

27 101 = 3761 - 30 * 122

28 107 = 3767 - 30 * 122

29  79 = 3769 - 30 * 123

30  89 = 3779 - 30 * 123

31 103 = 3793 - 30 * 123

32 107 = 3797 - 30 * 123

33 113 = 3803 - 30 * 123

34 131 = 3821 - 30 * 123

35 103 = 3823 - 30 * 124

36 113 = 3833 - 30 * 124

37 127 = 3847 - 30 * 124

38 131 = 3851 - 30 * 124

Later he send this beter one:

The function:

F(n) = Prime(n + 6393) - 30 * PrimePi(n + 18579) 

is prime for n = 1 to 39.

 

n F(n) = Prime(n + 6393) - 30 * PrimePi(n + 18579)

1 11

2 23

3 31

4 13

5 19

6 23

7 29

8 13

9 29

10 31

11 43

12 47

13 53

14 61

15 67

16 73

17 89

18 109

19 137

20 157

21 167

22 173

23 179

24 193

25 197

26 223

27 227

28 241

29 251

30 269

31 283

32 311

33 313

34 317

35 331

36 347

37 349

38 347

39 353

 

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Alain wrote:

Let be the function F(n)=[Prime(n+1)-PrimePi(n+5)]^2+Prime(n+1)-PrimePi(n+5)+41.

We get the following result n – F(n):

1 – 41  Prime
2 – 43  Prime
3 – 53  Prime
4 – 97  Prime
5 – 131  Prime
6 – 197  Prime
7 – 251  Prime
8 – 347  Prime
9 – 593  Prime
10 – 691  Prime
11 – 1033  Prime
12 – 1231  Prime
13 – 1373  Prime
14 – 1601  Prime
15 – 2111  Prime
16 – 2693  Prime
17 – 2903  Prime
18 – 3463  Prime
19 – 3947  Prime
20 – 4201  Prime
21 – 5011  Prime
22 – 5591  Prime
23 – 6521  Prime
24 - 7697=43*179

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Oscar wrote:

I checked for variations of the function given by Sebastián Martín Ruiz:
F(n) = Prime(n+a) + PrimePi(n+b)!
The given example is the case a=0, b=1.

F(n) is prime from n=1 to n=15:
F(n) = Prime(n+63307) + PrimePi(n+14)!
List of generated primes:
792031, 792037, 796361,
796387, 831683, 831697,
831707, 831731, 1154299,
1154311, 1154323, 1154327,
1154353, 1154369, 4420319.

F(n) is prime from n=1 to n=16:
F(n) = Prime(n+513368) + PrimePi(n+23)!
List of generated primes:
7944059, 7944133, 7944169, 7944191,
7944193, 11210117, 11210141, 47498147,
47498149, 47498189, 47498203, 47498207,
47498261, 486583063, 486583079, 486583081.

 

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