A220455 Number of ways to write n=x+y (x>0, y>0) with 3x-2, 3x+2 and 2xy+1 all prime

0, 0, 0, 1, 1, 2, 0, 2, 3, 2, 1, 2, 1, 1, 4, 4, 1, 2, 2, 3, 3, 2, 2, 5, 1, 4, 1, 1, 5, 4, 1, 2, 5, 5, 3, 8, 3, 6, 5, 5, 4, 4, 2, 4, 5, 3, 1, 8, 3, 4, 4, 1, 2, 8, 6, 3, 4, 5, 4, 4, 7, 1, 3, 6, 5, 7, 3, 3, 8, 2, 4, 5, 2, 6, 10, 7, 1, 5, 5, 6, 8, 6, 4, 5, 5, 7, 5, 4, 4, 11, 4, 5, 5, 5, 6, 6, 3, 1, 12, 8

Offset

1,6

Comments

Conjecture: a(n)>0 for all n>7.

This has been verified for n up to 10^8. It implies that there are infinitely many cousin primes.

Conjecture verified for n up to 10^9. - Mauro Fiorentini, Aug 06 2023

Zhi-Wei Sun also made some other similar conjectures, e.g., he conjectured that any integer n>17 can be written as x+y (x>0, y>0) with 2x-3, 2x+3 and 2xy+1 all prime, and each integer n>28 can be written as x+y (x>0, y>0) with 2x+1, 2y-1 and 2xy+1 all prime.

Both conjectures verified for n up to 10^9. - Mauro Fiorentini, Aug 06 2023

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.

Example

a(25)=1 since 25=13+12 with 3*13-2, 3*13+2 and 2*13*12+1=313 all prime.

Mathematica

a[n_]:=a[n]=Sum[If[PrimeQ[3k-2]==True&&PrimeQ[3k+2]==True&&PrimeQ[2k(n-k)+1]==True, 1, 0], {k, 1, n-1}]
Do[Print[n, " ", a[n]], {n, 1, 1000}]
apQ[{a_, b_}]:=AllTrue[{3a-2, 3a+2, 2a*b+1}, PrimeQ]; Table[Count[Flatten[ Permutations/@ IntegerPartitions[n, {2}], 1], _?(apQ[#]&)], {n, 100}] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jun 09 2018 *)

Crossrefs

Cf. A220431, A023200, A046132, A218867, A219055, A220419, A220413, A220272, A219842, A219864, A219923.

Sequence in context: A104513 A357618 A305610 * A208295 A285721 A214292

Adjacent sequences: A220452 A220453 A220454 * A220456 A220457 A220458

Keyword

nonn

Author

Zhi-Wei Sun, Dec 15 2012

Status

approved