Gaporial primes

[lukerichards]

Originally Posted by enzocreti

1 2

This is the vector of the prime gap records...who wants could find other gaporial primes with plus one ga#+1

I'll have a look when I get home, if others don't beat me to it.

[lukerichards]

ga1#+1 PRIME
ga2#+1 PRIME
ga3#+1 PRIME
ga4#+1 PRIME
ga5#+1
ga6#+1
ga7#+1 PRIME
ga8#+1
ga9#+1
ga10#+1
ga11#+1
ga12#+1
ga13#+1
ga14#+1
ga15#+1
ga16#+1
ga17#+1
ga18#+1
ga19#+1
ga20#+1
ga21#+1
ga22#+1
ga23#+1
ga24#+1
ga25#+1
ga26#+1
ga27#+1
ga28#+1
ga29#+1
ga30#+1
ga31#+1
ga32#+1
ga33#+1
ga34#+1
ga35#+1
ga36#+1
ga37#+1
ga38#+1
ga39#+1
ga40#+1
ga41#+1
ga42#+1
ga43#+1
ga44#+1
ga45#+1
ga46#+1
ga47#+1
ga48#+1
ga49#+1
ga50#+1
ga51#+1
ga52#+1
ga53#+1
ga54#+1
ga55#+1
ga56#+1
ga57#+1
ga58#+1
ga59#+1
ga60#+1
ga61#+1
ga62#+1
ga63#+1
ga64#+1
ga65#+1
ga66#+1
ga67#+1
ga68#+1
ga69#+1
ga70#+1
ga71#+1
ga72#+1
ga73#+1
ga74#+1
ga75#+1
ga76#+1
ga77#+1

[lukerichards]

ga1#+5 PRIME
ga2#+5 PRIME
ga3#+5 PRIME
ga4#+5 PRIME
ga5#+5
ga6#+5
ga7#+5
ga8#+5
ga9#+5
ga10#+5
ga11#+5
ga12#+5
ga13#+5
ga14#+5 PRIME*
ga15#+5
ga16#+5
ga17#+5
ga18#+5
ga19#+5
ga20#+5
ga21#+5
ga22#+5
ga23#+5
ga24#+5
ga25#+5
ga26#+5
ga27#+5
ga28#+5
ga29#+5
ga30#+5
ga31#+5
ga32#+5
ga33#+5
ga34#+5
ga35#+5
ga36#+5
ga37#+5
ga38#+5
ga39#+5
ga40#+5
ga41#+5
ga42#+5
ga43#+5
ga44#+5
ga45#+5
ga46#+5
ga47#+5
ga48#+5
ga49#+5
ga50#+5
ga51#+5
ga52#+5
ga53#+5
ga54#+5
ga55#+5
ga56#+5
ga57#+5
ga58#+5
ga59#+5
ga60#+5
ga61#+5
ga62#+5
ga63#+5
ga64#+5
ga65#+5
ga66#+5
ga67#+5
ga68#+5
ga69#+5
ga70#+5
ga71#+5
ga72#+5
ga73#+5
ga74#+5
ga75#+5
ga76#+5
ga77#+5

* this prime was added to the factordb by the act of searching for it.

All others are composite.

[Dylan14]

Originally Posted by Dylan14

Although it isn’t quite what you are considering, you may want to look at 081411 (partial product of all prime gaps) and consider finding a way to make primes from that sequence (not directly, of course, the only prime in that sequence is 2).

For example, using Mathematica we define the gaporial function as follows:

Gaporial[x_] := 
 Piecewise[{{1, 
    x == 0}, {Product[Prime[i + 1] - Prime[i], {i, 1, x}], 
    x > 0 && x \[Element] Integers}, {Product[
     Prime[i + 1] - Prime[i], {i, 1, Floor[x]}], 
    x > 0 && x \[NotElement] Integers}}]

which, when run with arguments 1 through 10, reproduces https://oeis.org/A081411. (This can be done with

Table[Gaporial[n], {n, 1, 10}]

)
Then, if we want to find primes related to this, we simply come up with a form. In the example I did gaporials+1:

If[PrimeQ[Gaporial[#] + 1], Print[#]] & /@ Range[100]

which gives the argument n such that gaporial(n)+1 is prime. This generates the following sequence: 1,2,3,4,7,10,15,17,19,35,57,59.

[enzocreti]

so the ga#+1 primes known are 3,7,43,967,5080977427

their sum is a prime!
3+4 is prime
7+4 is prime
43+4 is prime
967+4 is prime
5080977427+4 is prime

3+7+43+967+5080977427 is prime!

[enzocreti]

10*(5080978447)+1 is prime!

[Dylan14]

Originally Posted by Dylan14

For example, using Mathematica we define the gaporial function as follows:

Gaporial[x_] := 
 Piecewise[{{1, 
    x == 0}, {Product[Prime[i + 1] - Prime[i], {i, 1, x}], 
    x > 0 && x \[Element] Integers}, {Product[
     Prime[i + 1] - Prime[i], {i, 1, Floor[x]}], 
    x > 0 && x \[NotElement] Integers}}]

which, when run with arguments 1 through 10, reproduces https://oeis.org/A081411. (This can be done with

Table[Gaporial[n], {n, 1, 10}]

)
Then, if we want to find primes related to this, we simply come up with a form. In the example I did gaporials+1:

If[PrimeQ[Gaporial[#] + 1], Print[#]] & /@ Range[100]

which gives the argument n such that gaporial(n)+1 is prime. This generates the following sequence: 1,2,3,4,7,10,15,17,19,35,57,59.

I have continued the search of this form to n = 10000 using a mix of Mathematica (to generate the numbers) and pfgw (to test). Gaporial(n)+1 is (probably) prime for the following n:

1
2
3
4
7
10
15
17
19
35
57
59
121
142
204
296
307
400
410
480
573
591
730
904
1212
1436
1710
2178
2307
2390
4949
5949
6952