A219864 -id:A219864 - cp's OEIS frontend

A232194 Number of ways to write n = x + y (x, y > 0) with nx + y and ny - x both prime.

0, 0, 1, 1, 2, 1, 2, 2, 2, 3, 3, 3, 2, 3, 4, 2, 4, 2, 3, 1, 5, 4, 4, 1, 4, 3, 8, 3, 7, 2, 6, 3, 7, 4, 9, 3, 5, 4, 6, 3, 8, 4, 7, 5, 8, 3, 7, 4, 6, 3, 8, 3, 8, 2, 12, 4, 9, 4, 9, 4, 10, 3, 9, 7, 10, 5, 9, 4, 10, 4, 6, 5, 8, 3, 7, 5, 11, 7, 9, 8, 11, 5, 11, 8, 13, 4, 9, 5, 8, 7, 12, 6, 9, 5, 15, 7, 10, 5, 15, 10

Offset: 1

Zhi-Wei Sun, Nov 20 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 2. Also, a(n) = 1 only for n = 3, 4, 6, 20, 24.
(ii) Any positive integer n not among 1, 30, 54 can be written as x + y (x, y > 0) with n*x + y and n*y + x both prime.
(iii) Each integer n > 1 not equal to 8 can be expressed as x + y (x, y > 0) with n*x^2 + y (or x^4 + n*y) prime.
(iv) Any integer n > 5 can be written as p + q (q > 0) with p and n*q^2 + 1 both prime.
See also A232174 for a similar conjecture.

			a(3) = 1 since 3 = 1 + 2 with 3*1 + 2 = 3*2 - 1 = 5 prime.
a(4) = 1 since 4 = 1 + 3 with 4*1 + 3 = 7 and 4*3 - 1 = 11 both prime.
a(6) = 1 since 6 = 1 + 5 with 6*1 + 5 = 11 and 6*5 - 1 = 29 both prime.
a(20) = 1 since 20 = 9 + 11 with 20*9 + 11 = 191 and 20*11 - 9 = 211 both prime.
a(24) = 1 since 24*19 + 5 = 461 and 24*5 - 19 = 101 both 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. A000040, A218585, A218654, A219864, A220413, A227898, A227899, A232174, A232186.

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

A220572 Number of ways to write 2n-1=x+y (x,y>=0) with x^18+3*y^18 prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Dec 16 2012

nonn

Conjecture: a(n)>0 for every n=1,2,3,.... Moreover, any odd integer greater than 2092 can be written as x+y (x,y>0) with x-3, x+3 and x^18+3*y^18 all prime.
This has been verified for n up to 2*10^6.
Zhi-Wei Sun also made the following general conjecture: For each positive integer m, any sufficiently large odd integer n can be written as x+y (x,y>0) with x-3, x+3 and x^m+3*y^m all prime (and hence there are infinitely many primes in the form x^m+3*y^m). In particular, for m = 1, 2, 3, 4, 5, 6, 18 any odd integer greater than one can be written as x+y (x,y>0) with x^m+3*y^m prime, and for m =1, 2, 3 any odd integer n>15 can be written as x+y (x,y>0) with x-3, x+3 and x^m+3*y^m all prime.
Our computation suggests that for each m=7,...,20 any odd integer greater than 32, 10, 24, 30, 48, 36, 72, 146, 48, 48, 152, 2, 238, 84 respectively can be written as x+y (x,y>0) with x^m+3*y^m prime.

			a(3)=1 since 2*3-1=5=1+4 with 1^18+3*4^18=206158430209 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A220554, A036468, A220455, A220431, A218867, A219055, A220419, A220413, A220272, A219842, A219864, A219923.

a[n_]:=a[n]=Sum[If[PrimeQ[k^18+3*(2n-1-k)^18]==True,1,0],{k,0,2n-1}]
Do[Print[n," ",a[n]],{n,1,100}]

A230243 Number of primes p < n with 3p + 8 and (p-1)n + 1 both prime.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 1, 4, 2, 1, 4, 2, 2, 4, 2, 3, 2, 4, 3, 4, 4, 2, 2, 2, 1, 5, 3, 4, 3, 3, 2, 3, 4, 2, 2, 4, 2, 4, 4, 1, 5, 3, 2, 6, 4, 1, 5, 6, 3, 3, 5, 1, 5, 5, 2, 7, 5, 3, 4, 4, 3, 4, 6, 3, 4, 6, 4, 5, 6, 3, 7, 4, 2, 6, 1, 3, 5, 9, 3, 3, 7, 4, 3, 7, 1, 6, 5, 5, 5, 6, 3, 6, 7

Offset: 1

Zhi-Wei Sun, Oct 13 2013

nonn

Conjecture: a(n) > 0 for all n > 4.
This implies A. Murthy's conjecture (cf. A034693) that for any integer n > 1, there is a positive integer k < n such that k*n + 1 is prime.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Sep 21 2023

			a(8) = 1 since 8 = 3 + 5 with 3, 3*3+8 = 17, (3-1)*8+1 = 17 all prime.
a(17) = 1 since 17 = 7 + 10, and 7, 3*7+8 = 29, (7-1)*17+1 = 103 are 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. A034693, A085053, A086685, A023210, A219864, A230217, A230219, A230241.

a[n_]:=Sum[If[PrimeQ[3Prime[i]+8]&&PrimeQ[(Prime[i]-1)n+1],1,0],{i,1,PrimePi[n-1]}]
Table[a[n],{n,1,100}]

A231635 Number of ways to write n = x + y with 0 < x <= y such that lcm(x, y) + 1 is prime.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 3, 2, 1, 1, 4, 2, 6, 1, 3, 2, 8, 4, 5, 4, 3, 2, 7, 5, 6, 2, 3, 2, 8, 5, 10, 6, 3, 1, 8, 3, 9, 4, 4, 4, 14, 6, 16, 7, 7, 2, 12, 6, 8, 4, 5, 5, 21, 5, 8, 6, 4, 8, 11, 7, 12, 5, 6, 4, 10, 8, 22, 6, 10, 6, 17, 9, 23, 7, 11, 12, 18, 10, 19, 10, 10, 7, 23, 8, 15, 4, 7, 8, 14, 11, 19, 9, 2, 4, 11, 10, 35, 6, 10, 10

Offset: 1

Zhi-Wei Sun, Nov 12 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 1. Also, any integer n > 3 can be written as x + y (x, y > 0) with lcm(x, y) - 1 prime.
(ii) Each n = 2, 3, ... can be expressed as x + y (x, y > 0) with lcm(x, y)^2 + lcm(x, y) + 1 prime. Also, any integer n > 1 not equal to 10 can be written as x + y (x, y > 0) with lcm(x, y)^2 + 1 prime.
From Mauro Fiorentini, Aug 02 2023: (Start)
Both parts of conjecture (i) verified for n up to 10^9.
Both parts of conjecture (ii) verified for n up to 10^6. (End)

			a(9) = 1 since 9 = 3 + 6 with lcm(3, 6) + 1 = 7 prime.
a(10) = 1 since 10 = 4 + 6 with lcm(4, 6) + 1 = 13 prime.

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

Cf. A109909, A219842, A219864, A220272, A231201, A231516, A231555, A231561, A231633.

a[n_]:=Sum[If[PrimeQ[LCM[x,n-x]+1],1,0],{x,1,n/2}]
Table[a[n],{n,1,100}]

A231883 Number of ways to write n = x + y (x, y > 0) with x^2 + (n-2)*y^2 prime.

Original entry on oeis.org

0, 0, 2, 2, 2, 2, 4, 1, 5, 2, 5, 1, 4, 4, 3, 1, 7, 2, 3, 3, 6, 7, 3, 2, 6, 2, 9, 3, 8, 3, 10, 3, 5, 8, 8, 4, 7, 5, 13, 4, 12, 6, 7, 6, 8, 10, 14, 4, 17, 9, 9, 6, 9, 5, 8, 5, 9, 7, 12, 10, 11, 7, 11, 8, 12, 4, 13, 3, 22, 6, 16, 7, 14, 8, 10, 4, 14, 4, 17, 9, 16, 6, 12, 11, 14, 4, 21, 4, 21, 8, 18, 3, 11, 14, 23, 7, 22, 5, 23, 8

Offset: 1

Zhi-Wei Sun, Nov 21 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 2, and a(n) = 1 only for n = 8, 12, 16. Moreover, if m and n are positive integers with m >= max{2, n-1} and gcd(m, n+1) = 1, then x^2 + n*y^2 is prime for some positive integers x and y with x + y = m, except for the case m = n + 3 = 29.
(ii) Let m and n be integers greater than one with m >= (n-1)/2 and gcd(m, n-1) = 1. Then x + n*y is prime for some positive integers x and y with x + y = m.
(iii) Any integer n > 3 not equal to 12 or 16 can be written as x + y (x, y > 0) with (n-2)*x - y and (n-2)*x^2 + y^2 both prime.

			 a(8) = 1 since 8 = 5 + 3 with 5^2 + (8-2)*3^2 = 79 prime.
a(12) = 1 since 12 = 11 + 1 with 11^2 + (12-2)*1^2 = 131 prime.
a(16) = 1 since 16 = 15 + 1 with 15^2 + (16-2)*1^2 = 239 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. A000040, A036468, A219842, A219864, A220417, A232174, A232186, A232194.

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

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A232194 Number of ways to write n = x + y (x, y > 0) with n*x + y and n*y - x both prime.

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A220572 Number of ways to write 2n-1=x+y (x,y>=0) with x^18+3*y^18 prime.

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A230243 Number of primes p < n with 3*p + 8 and (p-1)*n + 1 both prime.

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A231635 Number of ways to write n = x + y with 0 < x <= y such that lcm(x, y) + 1 is prime.

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A231883 Number of ways to write n = x + y (x, y > 0) with x^2 + (n-2)*y^2 prime.

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A232194 Number of ways to write n = x + y (x, y > 0) with nx + y and ny - x both prime.

A230243 Number of primes p < n with 3p + 8 and (p-1)n + 1 both prime.