A219864 -id:A219864 - cp's OEIS frontend

A220413 Number of ways to write n=x+y (x>=0, y>=0) with x^3+2*y^3 prime.

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 3, 2, 3, 3, 3, 3, 2, 3, 4, 1, 4, 2, 3, 3, 3, 5, 5, 5, 3, 3, 5, 4, 4, 5, 6, 7, 4, 4, 5, 2, 6, 5, 5, 5, 4, 2, 4, 6, 4, 5, 4, 4, 8, 6, 5, 11, 6, 6, 8, 10, 5, 5, 5, 8, 6, 6, 11, 7, 5, 7, 9, 7, 6, 7, 8, 9, 6, 8, 10, 7, 11, 8, 7, 10, 9, 9, 6, 5, 7, 8, 13, 7, 9, 13, 13, 12, 9, 9

Offset: 1

Zhi-Wei Sun, Dec 13 2012

nonn

Conjecture: a(n)>0 for every n=1,2,3,... Moreover, any integer n>3 not among 7, 22, 31 can be written as p+q (q>0) with p and p^3+2*q^3 both prime.
We have verified this conjecture for n up to 10^8. D. R. Heath-Brown proved in 2001 that there are infinitely many primes in the form x^3+2*y^3, where x and y are positive integers.
Zhi-Wei Sun also made the following general conjecture: For each positive odd integer m, any sufficiently large integer n can be written as x+y (x>=0, y>=0) with x^m+2*y^m prime.
When m=1, this follows from Bertrand's postulate proved by Chebyshev in 1850. For m = 5, 7, 9, 11, 13, 15, 17, 19, it suffices to require that n is greater than 46, 69, 141, 274, 243, 189, 320, 454 respectively.

			a(9)=1 since 9=7+2 with 7^3+2*2^3=359 prime.
a(22)=1 since 22=1+21 with 1^3+2*21^3=18523 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
D. R. Heath-Brown, Primes represented by x^3 + 2y^3. Acta Mathematica 186 (2001), pp. 1-84.
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A220272, A219842, A219864, A219923.

a[n_]:=a[n]=Sum[If[PrimeQ[k^3+2(n-k)^3]==True,1,0],{k,0,n}]
Do[Print[n," ",a[n]],{n,1,100}]

A220272 Number of ways to write n=x^2+y (x>0, y>0) with 2xy-1 prime.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 2, 1, 1, 3, 2, 2, 2, 1, 1, 3, 3, 1, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 1, 1, 4, 3, 1, 2, 2, 2, 5, 3, 3, 3, 4, 3, 3, 1, 3, 3, 2, 2, 4, 4, 2, 6, 2, 2, 4, 4, 2, 3, 1, 2, 5, 4, 1, 3, 3, 3, 6, 2, 3, 5, 4, 3, 3, 3, 3, 6, 3, 2, 4, 2, 3, 4, 3, 2, 5, 3, 5, 2, 1, 1, 9, 4, 3, 4, 3, 5, 3, 2, 2, 2

Offset: 1

Zhi-Wei Sun, Dec 09 2012

nonn

Conjecture: a(n)>0 for all n>2.
This has been verified for n up to 3*10^9. The author observed that for each n=3,...,3*10^9 we may even require x<(log n)^2, but Jack Brennen found that for n=4630581798 we cannot require x<(log n)^2.
The author guessed that the conjecture can be slightly refined as follows: Any integer n>2 can be written as x^2+y with 2*x*y-1 prime, where x and y are positive integers with x<=y.
Zhi-Wei Sun also made the following general conjecture: If m is a positive integer and r is 1 or -1, then any sufficiently large integer n can be written as x^2+y (x>0, y>0) with m*x*y+r prime.
For example, for (m,r)=(1,-1),(1,1),(2,1),(3,-1),(3,1),(4,-1),(4,1),(5,-1),(5,1),(6,-1),(6,1), it suffices to require that n is greater than 12782, 15372, 488, 5948, 2558, 92, 822, 21702, 6164, 777, 952 respectively.

			a(18)=1 since 18=3^2+9 with 2*3*9-1=53 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..20000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.
Zhi-Wei Sun, Re: A curious conjecture on primes, a message to Number Theory List, Dec. 12, 2012.

Cf. A219842, A219864, A219923.

a[n_]:=a[n]=Sum[If[PrimeQ[2k(n-k^2)-1]==True,1,0],{k,1,Sqrt[n]}]
Do[Print[n," ",a[n]],{n,1,100}]

A199920 Number of ways to write n = p+k with p, p+6, 6k-1 and 6k+1 all prime.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 1, 2, 0, 3, 1, 3, 2, 2, 2, 3, 2, 2, 1, 2, 3, 3, 3, 1, 1, 3, 2, 4, 1, 2, 2, 3, 3, 3, 2, 4, 2, 4, 3, 3, 5, 3, 3, 3, 3, 4, 5, 3, 3, 3, 3, 5, 4, 4, 3, 4, 3, 3, 2, 3, 6, 5, 4, 2, 1, 3, 5, 5, 5, 2, 2, 3, 5, 3, 5, 4, 5, 2, 3, 2, 5, 5, 6, 4, 2, 3, 3, 4, 3, 3, 5, 4, 3, 1, 1, 4, 5, 7

Offset: 1

Zhi-Wei Sun, Dec 22 2012

Conjecture: a(n)>0 for all n>11.
This implies that there are infinitely many twin primes and also infinitely many sexy primes. It has been verified for n up to 10^9. See also A199800 for a weaker version of this conjecture.
Zhi-Wei Sun also conjectured that any integer n>6 not equal to 319 can be written as p+k with p, p+6, 3k-2+(n mod 2) and 3k+2-(n mod 2) all prime.

			a(21)=1 since 21=11+10 with 11, 11+6, 6*10-1 and 6*10+1 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..50000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A001359, A006512, A023201, A199800, A219055, A219864, A219923, A002375.

a[n_]:=a[n]=Sum[If[PrimeQ[Prime[k]+6]==True&&PrimeQ[6(n-Prime[k])-1]==True&&PrimeQ[6(n-Prime[k])+1]==True,1,0],{k,1,PrimePi[n]}]
Do[Print[n," ",a[n]],{n,1,100}]
Table[Count[Table[{n-i,i},{i,n-1}],?(AllTrue[{#[[1]],#[[1]]+6,6#[[2]]-1,6#[[2]]+1},PrimeQ]&)],{n,100}] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, May 19 2015 *)

a(n)=my(s,p=2,q=3); forprime(r=5,n+5, if(r-p==6 && isprime(6*n-6*p-1) && isprime(6*n-6*p+1), s++); if(r-q==6 && isprime(6*n-6*q-1) && isprime(6*n-6*q+1), s++); p=q; q=r); s \\ Charles R Greathouse IV, Jul 31 2016

A219923 Number of ways to write n=x+y (x>0, y>0) with x-1, x+1 and 2xy+1 all prime.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 1, 1, 1, 1, 3, 2, 0, 1, 2, 2, 3, 2, 1, 0, 2, 2, 0, 1, 3, 2, 2, 1, 3, 4, 2, 2, 3, 0, 4, 3, 3, 1, 1, 3, 0, 3, 2, 1, 1, 3, 3, 1, 1, 5, 3, 1, 2, 1, 3, 3, 5, 3, 1, 2, 4, 3, 3, 2, 4, 3, 2, 2, 0, 3, 5, 4, 1, 3, 6, 2, 6, 2, 2, 4, 5, 5, 2, 3, 3, 4, 1, 2, 0, 1, 4, 2, 4, 1, 6, 6

Offset: 1

Zhi-Wei Sun, Dec 01 2012

nonn

Conjecture: a(n)>0 for all n>623.
This has been verified for n up to 10^8.
Zhi-Wei Sun made the following general conjecture: For each nonnegative integer m, any sufficiently large integer n can be written as x+y (x>0, y>0) with x-m, x+m and 2*x*y+1 all prime.
For example, when m = 2, 3, 4, 5 it suffices to require that n is greater than 28, 151, 357, 199 respectively.
Sun also conjectured that for each m=0,1,2,... any sufficiently large integer n with m or n odd can be written as x+y (x>0, y>0) with x-m, x+m and x*y-1 all prime.
For example, in the case m=1 it suffices to require that n is greater than 4 and not among 40, 125, 155, 180, 470, 1275, 2185, 3875; when m=2 it suffices to require that n is odd, greater than 7, and different from 13.

			a(11)=1 since 11=6+5 with 6-1, 6+1 and 2*6*5+1=61 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A219864, A219842.

a[n_]:=a[n]=Sum[If[PrimeQ[Prime[k]+2]==True&&PrimeQ[2(Prime[k]+1)(n-Prime[k]-1)+1]==True,1,0],{k,1,PrimePi[n-1]}]
Do[Print[n," ",a[n]],{n,1,10000}]

A232174 Number of ways to write n = x + y (x, y > 0) with x + ny and x^2 + ny^2 both prime.

Original entry on oeis.org

0, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 2, 5, 1, 4, 3, 2, 2, 1, 1, 2, 5, 4, 1, 7, 2, 4, 4, 6, 2, 5, 1, 4, 3, 5, 2, 8, 2, 6, 3, 3, 3, 5, 2, 5, 4, 7, 5, 7, 3, 5, 3, 3, 1, 11, 4, 7, 6, 5, 2, 4, 3, 8, 5, 6, 1, 14, 1, 6, 7, 6, 6, 8, 3, 6, 7, 7, 5, 9, 3, 3, 5, 7, 7, 15, 5, 6, 5, 2, 5, 15, 6, 12, 8, 7, 3, 15, 8, 10, 5

Offset: 1

Zhi-Wei Sun, Nov 19 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 1. Also, a(n) = 1 only for n = 2, 5, 8, 14, 19, 20, 24, 32, 54, 68, 101, 168.
(ii) Every n = 3, 4, ... can be written as x + y (x, y > 0) with x*n + y and x*n - y both prime.
(iii) Any integer n > 2 can be written as P + q (q > 0) with p and p + n*q both prime. Also, any integer n > 7 can be written as p + q (q > 0) with p and n*q - p both prime.
In a paper published in 2017, the author announced a USD $200 prize for the first solution to his conjecture that a(n) > 0 for all n > 1. - Zhi-Wei Sun, Dec 03 2017

			a(2) = 1 since 2 = 1 + 1 with 1 + 2*1 = 1^2 + 2*1^2 = 3 prime.
a(5) = 1 since 5 = 3 + 2 with 3 + 5*2 = 13 and 3^2 + 5*2^2 = 29 both prime.
a(8) = 1 since 8 = 5 + 3 with 5 + 8*3 = 29 and 5^2 + 8*3^2 = 97 both prime.
a(14) = 1 since 14 = 9 + 5 with 9 + 14*5 = 79 and 9^2 + 14*5^2 = 431 both prime.
a(19) = 1 since 19 = 13 + 6 with 13 + 19*6 = 127 and 13^2 + 19*6^2 = 853 both prime.
a(20) = 1 since 20 = 11 + 9 with 11 + 20*9 = 191 and 11^2 + 20*9^2 = 1741 both prime.
a(24) = 1 since 24 = 5 + 19 with 5 + 24*19 = 461 and 5^2 + 24*19^2 = 8689 both prime.
a(32) = 1 since 32 = 23 + 9 with 23 + 32*9 = 311 and 23^2 + 32*9^2 = 3121 both prime.
a(54) = 1 since 54 = 35 + 19 with 35 + 54*19 = 1061 and 35^2 + 54*19^2 = 20719 both prime.
a(68) = 1 since 68 = 45 + 23 with 45 + 68*23 = 1609 and 45^2 + 68*23^2 = 37997 both prime.
a(101) = 1 since 101 = 98 + 3 with 98 + 101*3 = 401 and 98^2 + 101*3^2 = 10513 both prime.
a(168) = 1 since 168 = 125 + 43 with 125 + 168*43 = 7349 and 125^2 + 168*43^2 = 326257 both prime.

D. A. Cox, Primes of the Form x^2 + n*y^2, John Wiley & Sons, 1989.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016, Springer Proc. in Math. & Stat., Vol. 220, Springer, New York, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT].)

Cf. A000040, A036468, A219842, A219864, A220413.

a[n_]:=Sum[If[PrimeQ[k+n(n-k)]&&PrimeQ[k^2+n(n-k)^2],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A230351 Number of ordered ways to write n = p + q (q > 0) with p, 2p^2 - 1 and 2q^2 - 1 all prime.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 1, 1, 3, 3, 2, 1, 4, 3, 4, 2, 4, 3, 4, 5, 4, 2, 3, 6, 3, 3, 3, 5, 2, 3, 3, 3, 1, 2, 4, 2, 2, 3, 3, 1, 5, 2, 3, 3, 7, 3, 5, 4, 6, 3, 5, 6, 5, 5, 3, 6, 2, 5, 5, 3, 4, 5, 6, 2, 6, 6, 5, 1, 5, 3, 3, 3, 2, 2, 5, 6, 5, 1, 5, 6, 4, 4, 6, 6, 1, 5, 5, 4, 3, 4, 3, 3, 6, 5, 4, 1, 5, 7, 2, 4

Offset: 1

Zhi-Wei Sun, Oct 16 2013

nonn

Conjecture: a(n) > 0 for all n > 3.
We have verified this for n up to 2*10^7.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Aug 07 2023

			a(7) = 1 since 7 = 3 + 4 with 3, 2*3^2 - 1 = 17, 2*4^2 - 1 = 31 all prime.
a(40) = 1 since 40 = 2 + 38, and 2, 2*2^2 - 1 = 7 , 2*38^2 - 1 = 2887 are all prime.
a(68) = 1 since 68 = 43 + 25, and all the three numbers 43, 2*43^2 - 1 = 3697 and 2*25^2 - 1 = 1249 are prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A000040, A066049, A106483, A219864, A230252, A230254, A230261.

a[n_]:=Sum[If[PrimeQ[2Prime[i]^2-1]&&PrimeQ[2(n-Prime[i])^2-1],1,0],{i,1,PrimePi[n-1]}]
Table[a[n],{n,1,100}]

A220419 Number of ways to write n=x+y (x>0, y>0) with 2x+1, 2y-1 and x^3+2y^3 all prime.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 3, 1, 1, 0, 2, 1, 2, 1, 2, 1, 2, 0, 1, 1, 0, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 4, 2, 1, 1, 1, 3, 2, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 3, 3, 1, 4, 1, 1, 0, 4, 2, 2, 3, 0, 1, 3, 2, 2, 1, 0, 5, 2, 0, 0, 1, 2, 2, 2, 2, 1, 2, 2, 3, 3, 2, 0, 1, 0, 2, 2, 4, 3, 2, 1, 3, 4, 2, 3

Offset: 1

Zhi-Wei Sun, Dec 14 2012

nonn

Conjecture: a(n)>0 for all n>527.
This has been verified for n up to 2*10^7. It implies the Goldbach conjecture since 2(x+y)=(2x+1)+(2y-1).
Zhi-Wei Sun also made the following similar conjectures:
(1) Each integer n>1544 can be written as x+y (x>0, y>0) with 2x-1, 2y+1 and x^3+2y^3 all prime.
(2) Any odd number n>2060 can be written as 2p+q with p, q and p^3+2((q-1)/2)^3 all prime.
(3) Every integer n>25537 can be written as p+q (q>0) with p, p-6, p+6 and p^3+2q^3 all prime.
(4) Any even number n>1194 can be written as x+y (x>0, y>0) with x^3+2y^3 and 2x^3+y^3 both prime.
(5) Each integer n>3662 can be written as x+y (x>0, y>0) with 3(xy)^3-1 and 3(xy)^3+1 both prime.
(6) Any integer n>22 can be written as x+y (x>0, y>0) with (xy)^4+1 prime. Also, any integer n>7425 can be written as x+y (x>0, y>0) with 2(xy)^4-1 and 2(xy)^4+1 both prime.
(7) Every odd integer n>1 can be written as x+y (x>0, y>0) with x^4+y^2 prime. Moreover, any odd number n>15050 can be written as p+2q with p, q and p^4+(2q)^2 all prime.
Conjectures (1) to (7) verified up to 10^6. - Mauro Fiorentini, Sep 22 2023

			a(25)=1 since 25=3+22 with 2*3+1, 2*22-1 and 3^3+2*22^3=21323 all prime.
a(26)=1 since 26=11+15 with 2*11+1, 2*15-1 and 11^3+2*15^3=8081 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..20000
D. R. Heath-Brown, Primes represented by x^3 + 2y^3. Acta Mathematica 186 (2001), pp. 1-84.
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A220413, A173587, A220272, A219842, A219864, A219923.

a[n_]:=a[n]=Sum[If[PrimeQ[2k+1]==True&&PrimeQ[2(n-k)-1]==True&&PrimeQ[k^3+2(n-k)^3]==True,1,0],{k,1,n-1}]
Do[Print[n," ",a[n]],{n,1,1000}]

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

Original entry on oeis.org

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

Zhi-Wei Sun, Dec 15 2012

nonn

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

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

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.

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

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 *)

A230040 Number of ways to write n = x + y + z with y <= z such that all the five numbers 6x-1, 6y-1, 6z-1, 6x*y-1 and 6xz-1 are Sophie Germain primes.

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 3, 1, 3, 4, 5, 2, 1, 1, 3, 4, 4, 3, 4, 6, 5, 2, 2, 6, 5, 1, 2, 4, 2, 2, 3, 6, 5, 7, 6, 2, 3, 4, 4, 2, 3, 5, 1, 4, 7, 4, 6, 3, 9, 4, 2, 5, 4, 3, 9, 2, 4, 3, 6, 3, 5, 8, 8, 5, 8, 6, 2, 4, 3, 4, 1, 6, 4, 3, 8, 8, 6, 6, 9, 11, 2, 4, 2, 8, 3, 4, 6, 10, 5, 11, 7, 8, 6, 10, 4, 1, 3, 1, 3, 3

Offset: 1

Zhi-Wei Sun, Oct 06 2013

nonn

Conjecture: a(n) > 0 for all n > 2.
This implies that 6*n-3 with n > 2 can be expressed as a sum of three Sophie Germain primes (i.e., those primes p with 2*p+1 also prime).
We have verified the conjecture for n up to 10^8. Note that any Sophie Germain prime p > 3 has the form 6*k-1.

			a(4) = 2, since 4 = 1 + 1 + 2 = 2 + 1 + 1, and 6*1-1=5 and 6*2-1=11 are Sophie Germain primes.
a(26) = 1, since 26 = 15 + 2 + 9, and all the five numbers 6*15-1=89, 6*2-1=11, 6*9-1=53, 6*15*2-1=179 and 6*15*9=809 are Sophie Germain primes.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Cf. A005384, A219842, A219864, A227938, A229969, A229974, A230037.

SQ[n_]:=PrimeQ[n]&&PrimeQ[2n+1]
a[n_]:=Sum[If[SQ[6i-1]&&SQ[6j-1]&&SQ[6(n-i-j)-1]&&SQ[6i*j-1]&&SQ[6*i(n-i-j)-1],1,0],{i,1,n-2},{j,1,(n-i)/2}]
Table[a[n],{n,1,100}]

A230252 Number of ways to write n = x + y (x, y > 0) with 2*x + 1, x^2 + x + 1 and y^2 + y + 1 all prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Oct 13 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 1.
(ii) Any integer n > 3 can be written as p + q with p, 2*p - 3 and q^2 + q + 1 all prime. Also, each integer n > 3 not equal to 30 can be expressed as p + q with p, q^2 + q - 1 and q^2 + q + 1 all prime.
(iii) Any integer n > 1 can be written as x + y (x, y > 0) with x^2 + 1 (or 4*x^2+1) and y^2 + y + 1 (or 4*y^2 + 1) both prime.
(iv) Each integer n > 3 can be expressed as p + q (q > 0) with p, 2*p - 3 and 4*q^2 + 1 all prime.
(v) Any even number greater than 4 can be written as p + q with p, q and p^2 + 4 (or p^2 - 2) all prime. Also, each even number greater than 2 and not equal to 122 can be expressed as p + q with p, q and (p-1)^2 + 1 all prime.
We have verified the first part for n up to 10^8.

			a(5) = 2 since 5 = 2 + 3 = 3 + 2, and 2*2+1 = 5, 2*3+1 = 7, 2^2+2+1 = 7, 3^2+3+1 = 13 are all prime.
a(31) = 1 since 31 = 14 + 17, and 2*14+1 = 29, 14^2+14+1 = 211 and 17^2+17+1 = 307 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Cf. A002383, A002384, A002375, A219864, A227908, A227909, A230230, A230241, A230254.

a[n_]:=Sum[If[PrimeQ[2i+1]&&PrimeQ[i^2+i+1]&&PrimeQ[(n-i)^2+n-i+1],1,0],{i,1,n-1}]
Table[a[n],{n,1,100}]

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A220413 Number of ways to write n=x+y (x>=0, y>=0) with x^3+2*y^3 prime.

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A220272 Number of ways to write n=x^2+y (x>0, y>0) with 2*x*y-1 prime.

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A199920 Number of ways to write n = p+k with p, p+6, 6k-1 and 6k+1 all prime.

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A219923 Number of ways to write n=x+y (x>0, y>0) with x-1, x+1 and 2*x*y+1 all prime.

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A232174 Number of ways to write n = x + y (x, y > 0) with x + n*y and x^2 + n*y^2 both prime.

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A230351 Number of ordered ways to write n = p + q (q > 0) with p, 2*p^2 - 1 and 2*q^2 - 1 all prime.

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A220419 Number of ways to write n=x+y (x>0, y>0) with 2x+1, 2y-1 and x^3+2y^3 all prime.

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

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A220272 Number of ways to write n=x^2+y (x>0, y>0) with 2xy-1 prime.

A219923 Number of ways to write n=x+y (x>0, y>0) with x-1, x+1 and 2xy+1 all prime.

A232174 Number of ways to write n = x + y (x, y > 0) with x + ny and x^2 + ny^2 both prime.

A230351 Number of ordered ways to write n = p + q (q > 0) with p, 2p^2 - 1 and 2q^2 - 1 all prime.

A230040 Number of ways to write n = x + y + z with y <= z such that all the five numbers 6x-1, 6y-1, 6z-1, 6x*y-1 and 6xz-1 are Sophie Germain primes.