Puzzle 1217 6*(Prime(n+1)-Prime(n))-7

During the week 18-24 April, 2025, contributions came from Giorgos Kalogeropoilos, Michael Branicky, Grant Bell, Simon Cavegn, Gennady Gusev.

Comment: Only Gennady Gusev found a Functions with 20 succesive primes and these primes are all distinct!. See in red below.

F(n) is prime for n=2 to 188

29,29,17,29,17,29,17,5,29,5,17,29,17,5,5,29,5,17,29,5,17,5,7,17,29,17,29,17,43,17,5,29,19,29,5,5,17,5,5,29,
19,29,17,29,31,31,17,29,17,5,29,19,5,5,5,29,5,17,29,19,43,17,29,17,43,5,19,29,17,5,7,5,5,17,5,7,17,7,19,29,
19,29,5,17,5,7,17,29,17,31,7,17,7,17,5,31,29,67,5,19,5,5,29,5,19,5,5,29,5,5,17,29,31,19,29,17,5,5,29,31,17,
5,7,19,7,19,7,5,5,17,7,5,17,7,17,43,19,31,29,19,29,17,29,19,43,17,29,17,43,17,29,17,79,17,7,19,7,17,5,5,43,
17,5,5,7,5,31,17,5,29,19,29,5,19,29,19,29,5,67,17,29,17,5,5,7,5,5

9 distinct primes {5, 7, 17, 19, 29, 31, 43, 67, 79

Searching F(n) = a*(prime(n+1) - prime(n)) - b from n = s, I found

F(n) = 3*(prime(n+1) - prime(n)) - 1 gives 46 successive primes (7 unique) from n=1..46

* 46 primes: [2, 2, 5, 5, 11, 5, 11, 5, 11, 17, 5, 17, 11, 5, 11, 17, 17, 5, 17, 11, 5, 17, 11, 17, 23, 11, 5, 11, 5, 11, 41, 11, 17, 5, 29, 5, 17, 17, 11, 17, 17, 5, 29, 5, 11, 5]
* 7 unique: {2, 5, 11, 17, 23, 29, 41}

F(n) = 120*(prime(n+1) - prime(n)) + 91 gives 77 successive primes (11 unique) from n=791..867

* 77 primes: [811, 811, 811, 1291, 331, 1291, 1531, 1051, 1291, 331, 1291, 1051, 1531, 1291, 2971, 331, 571, 1051, 811, 571, 1051, 2251, 1291, 811, 811, 331, 811, 1291, 1531, 331, 1291, 811, 811, 811, 1051, 811, 1291, 811, 331, 811, 811, 811, 1291, 1051, 2971, 811, 2731, 331, 2251, 571, 1051, 1291, 3691, 1051, 2251, 571, 331, 1291, 811, 331, 811, 571, 2251, 1051, 1531, 2251, 2011, 811, 331, 1531, 811, 1291, 331, 1291, 331, 811, 1291]
* 11 unique: {331, 571, 811, 1051, 1291, 1531, 2011, 2251, 2731, 2971, 3691}

F(n) = 105*(prime(n+1) - prime(n)) - 11 gives 188 successive primes (9 unique) from n=2..189

* 188 primes: [199, 199, 199, 409, 199, 409, 199, 409, 619, 199, 619, 409, 199, 409, 619, 619, 199, 619, 409, 199, 619, 409, 619, 829, 409, 199, 409, 199, 409, 1459, 409, 619, 199, 1039, 199, 619, 619, 409, 619, 619, 199, 1039, 199, 409, 199, 1249, 1249, 409, 199, 409, 619, 199, 1039, 619, 619, 619, 199, 619, 409, 199, 1039, 1459, 409, 199, 409, 1459, 619, 1039, 199, 409, 619, 829, 619, 619, 409, 619, 829, 409, 829, 1039, 199, 1039, 199, 619, 409, 619, 829, 409, 199, 409, 1249, 829, 409, 829, 409, 619, 1249, 199, 1879, 619, 1039, 619, 619, 199, 619, 1039, 619, 619, 199, 619, 619, 409, 199, 1249, 1039, 199, 409, 619, 619, 199, 1249, 409, 619, 829, 1039, 829, 1039, 829, 619, 619, 409, 829, 619, 409, 829, 409, 1459, 1039, 1249, 199, 1039, 199, 409, 199, 1039, 1459, 409, 199, 409, 1459, 409, 199, 409, 2089, 409, 829, 1039, 829, 409, 619, 619, 1459, 409, 619, 619, 829, 619, 1249, 409, 619, 199, 1039, 199, 619, 1039, 199, 1039, 199, 619, 1879, 409, 199, 409, 619, 619, 829, 619, 619])
* 9 unique: {199, 409, 619, 829, 1039, 1249, 1459, 1879, 2089}

Searching F(n) = a*(b*prime(n+1) - c*prime(n)) - d, from n = s, I found

F(n) = (-30*prime(n+1) + 31*prime(n)) + 30 gives 25 successive primes (23 unique) from n=76..100

* 25 primes: [233, 233, 179, 307, 191, 139, 389, 151, 401, 283, 349, 293, 239, 367, 431, 373, 137, 269, 397, 281, 409, 353, 179, 491, 13]

* 23 unique: [13, 137, 139, 151, 179, 191, 233, 239, 269, 281, 283, 293, 307, 349, 353, 367, 373, 389, 397, 401, 409, 431, 491]

Since a = b, the formula reduces to the simpler 30 * (P(n+1) - P(n)) + 71

F(0) is `101`
F(1) is `131`
F(2) is `131`
F(3) is `191`
F(4) is `131`
...
F(28) is `191`
F(29) is `491`
F(30) is `191`
F(31) is `251`
F(32) is `131`

More solutions include:

a = b = 9228, c = -3055 => n < [0, 29), F(n) < {6157, 15401, 33857, 52313, 70769}

a = b = 24030, c = -8479 => n < [0, 45), F(n) < {15551, 39581, 87641, 135701, 183761, 327941, 231821}

Longest sequences I could find where `a` and `b` were distinct I found particularly interesting:

a = 43, b = -2, c = -120

F(0) is `13`
F(1) is `101`
F(2) is `191`
F(3) is `367`
F(4) is `461`

a = -43, b = -58, c = 114

F(0) is `101`
F(1) is `73`
F(2) is `103`
F(3) is `47`
F(4) is `193`
F(5) is `137`
F(6) is `283`
F(7) is `227`

a = -21, b = -126, c = -50 => n < [0, 11)

a = 120, b = 30, c = -49 => n = [0, 13)

a = -15, b = -30, c = 14 => n = [0, 14) This is the first one where a != b where repeats of F(n) occur!

3*prime(n+1) - 3*prime(n) - 1
n=0..44
2,5,5,11,5,11,5,11,17,5,17,11,5,11,17,17,5,17,11,5,17,11,17,23,11,5,11,5,11,41,11,17,5,29,5,17,17,11,17,17,5,29,5,11,5
Distinct:2,5,11,17,23,29,41

if negative primes were allowed:

-6*prime(n+1) + 6*prime(n) + 65
n=0..152
59,53,53,41,53,41,53,41,29,53,29,41,53,41,29,29,53,29,41,53,29,41,29,17,41,53,41,53,41,-19,41,29,53,5,53,29,29,41,29,29,53,5,53,41,53,-7,-7,41,53,41,29,53,5,29,29,29,53,29,41,53,5,-19,41,53,41,-19,29,5,53,41,29,17,29,29,41,29,17,41,17,5,53,5,53,29,41,29,17,41,53,41,-7,17,41,17,41,29,-7,53,-43,29,5,29,29,53,29,5,29,29,53,29,29,41,53,-7,5,53,41,29,29,53,-7,41,29,17,5,17,5,17,29,29,41,17,29,41,17,41,-19,5,-7,53,5,53,41,53,5,-19,41,53,41,-19,41,53,41
Distinct:-43,-19,-7,5,17,29,41,53,59

-30*prime(n+1) + 30*prime(n) + 187
n=0..152
157,127,127,67,127,67,127,67,7,127,7,67,127,67,7,7,127,7,67,127,7,67,7,-53,67,127,67,127,67,-233,67,7,127,-113,127,7,7,67,7,7,127,-113,127,67,127,-173,-173,67,127,67,7,127,-113,7,7,7,127,7,67,127,-113,-233,67,127,67,-233,7,-113,127,67,7,-53,7,7,67,7,-53,67,-53,-113,127,-113,127,7,67,7,-53,67,127,67,-173,-53,67,-53,67,7,-173,127,-353,7,-113,7,7,127,7,-113,7,7,127,7,7,67,127,-173,-113,127,67,7,7,127,-173,67,7,-53,-113,-53,-113,-53,7,7,67,-53,7,67,-53,67,-233,-113,-173,127,-113,127,67,127,-113,-233,67,127,67,-233,67,127,67
Distinct:-353,-233,-173,-113,-53,7,67,127,157

If we write F(n) as a*prime(n+1) + b*prime(n) + c, then

For a,b,c: -93, 105, 35 and n=29, 30, ...48, we have exactly 20 primes and all are different:

For a,b,c: 31, -30, -180 and n=110, 111...135, we get 26 primes and 24 of them are different:

607 613 557 499 811 761 523 587 653 659 541 853 617 683 751 821 769 839 787 733 739 683 811 757 701 829 (The repetitive ones are highlighted in bold).

For a,b,c: 105, -105, -11 and n=2,3,... 188, we get 187 primes, but only 9 different primes

199 199 409 199 409 199 409 619 199 619 409 199 409 619 619 199 619 409 199 619 409 619 829 409 199 409 199 409 1459 409 619 199 1039 199 619 619 409 619 619 199 1039 199 409 199 1249 1249 409 199 409 619 199 1039 619 619 619 199 619 409 199 1039 1459 409 199 409 1459 619 1039 199 409 619 829 619 619 409 619 829 409 829 1039 199 1039 199 619 409 619 829 409 199 409 1249 829 409 829 409 619 1249 199 1879 619 1039 619 619 199 619 1039 619 619 199 619 619 409 199 1249 1039 199 409 619 619 199 1249 409 619 829 1039 829 1039 829 619 619 409 829 619 409 829 409 1459 1039 1249 199 1039 199 409 199 1039 1459 409 199 409 1459 409 199 409 2089 409 829 1039 829 409 619 619 1459 409 619 619 829 619 1249 409 619 199 1039 199 619 1039 199 1039 199 619 1879 409 199 409 619 619 829 619 619

different primes : 199 409 619 829 1459 1039 1249 1879 2089.

I found the formula abs(-45*[p(n+1)-p(n)]+533) was prime for n=2..281 (a sequence of 280 n values) and produced the following twelve unique primes {7, 83, 97, 367, 997, 443, 263, 353, 457, 173, 547, 277}.

Once I was focused on functions of the prime gap p(n+1)-p(n), I realized we could employ primes in arithmetic progressions. The first known length 27 PAP is 224584605939537911+81292139*23#*n which is prime for n=0..26 (where 23# stands for the product of primes <= 23) (due to Rob Gahan and PrimeGrid in 2019). For n from 2 to 30801, I get a list of primes gaps of {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 76, 78, 82, 86}, I subtract 42, take the absolute value, and divide by two to get arguments (for the PAP) of [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22].

So for 2<=n<=30801, the formula 224584605939537911+81292139*23#*( 1/2*abs(p(n+1)-p(n)-42) ) produces 22 distinct primes. Those distinct primes are {224584605939537911, 587298537898516511, 569162841300567581, 333398785527231491, 369670178723129351, 260855999135435771, 623569931094414371, 351534482125180421, 442212965114925071, 551027144702618651, 532891448104669721, 478484358310822931, 405941571919027211, 514755751506720791, 297127392331333631, 496620054908771861, 424077268516976141, 242720302537486841, 278991695733384701, 387805875321078281, 460348661712874001, 315263088929282561}.

One might find other sequences of primes where the gaps are "densely packed" and get a longer sequence for the n values, rather than starting with the small primes. TBA.