A231557 -id:A231557 - cp's OEIS frontend

A231633 Number of ways to write n = x + y (x, y > 0) with x^2 * y - 1 prime.

0, 0, 1, 2, 3, 1, 3, 2, 5, 2, 4, 2, 7, 2, 5, 3, 5, 3, 10, 4, 5, 3, 8, 3, 14, 6, 5, 4, 11, 5, 11, 3, 11, 9, 4, 5, 10, 5, 11, 9, 12, 3, 19, 7, 11, 6, 12, 9, 11, 7, 17, 7, 13, 5, 22, 3, 3, 15, 16, 5, 25, 4, 9, 11, 13, 5, 19, 6, 22, 6, 11, 6, 39, 6, 24, 7, 7, 6, 25, 8, 21, 11, 24, 7, 31, 7, 19, 11, 33, 10, 14, 8, 15, 27, 18, 9, 21, 4, 27, 9

Offset: 1

Zhi-Wei Sun, Nov 12 2013

nonn

Conjectures:
(i) a(n) > 0 for all n > 2. Also, any integer n > 4 can be written as x + y (x, y > 0) with x^2 * y + 1 prime.
(ii) Each n = 2, 3, ... can be expressed as x + y (x, y > 0) with (x*y)^2 + x*y + 1 prime.
(iii) Also, any integer n > 2 can be written as x + y (x, y > 0) with 2*(x*y)^2 - 1 (or (x*y)^2 + x*y - 1) prime.
From Mauro Fiorentini, Jul 31 2023: (Start)
Both parts of conjecture (i) verified for n up to 10^9.
Conjecture (ii) and both parts of conjecture (iii) verified for n up to 10^7. (End)

			a(6) = 1 since 6 = 4 + 2 with 4^2*2 - 1 = 31 prime.

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

Cf. A000040, A000290, A109909, A219842, A219864, A220272, A228424, A231201, A231155, A231561, A231557, A231631.

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

A231776 Least positive integer k <= n with (2^k + k) * n - 1 prime, or 0 if such a number k does not exist.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 6, 2, 10, 1, 2, 1, 2, 1, 2, 1, 4, 2, 2, 1, 2, 8, 6, 1, 2, 1, 4, 2, 2, 1, 8, 1, 4, 1, 2, 2, 14, 2, 2, 1, 2, 1, 2, 6, 2, 1, 4, 2, 2, 3, 8, 1, 6, 1, 2, 1, 8, 5, 4, 1, 2, 1, 2, 6, 42, 2, 6, 2, 4, 2, 2, 1, 2, 1, 4, 1, 4, 2, 8, 1, 2, 1, 2, 1, 6, 1, 8, 20, 2, 1, 2, 6, 10, 1, 2, 2

Offset: 1

Zhi-Wei Sun, Nov 13 2013

nonn

We find that 75011 is the only value of n <= 10^5 with a(n) = 0. The least positive integer k with (2^k + k)*75011 - 1 prime is 81152.

			a(3) = 2 since (2^1 + 1) * 3 - 1 = 8 is not prime, but (2^2 + 2) * 3 - 1 = 17 is prime.

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

Cf. A000040, A000079, A231201, A231557, A231561, A231725.

Do[Do[If[PrimeQ[(2^k+k)*n-1],Print[n," ",k];Goto[aa]],{k,1,n}]; Print[n," ",0];Label[aa];Continue,{n,1,100}]
lpi[n_]:=Module[{k=1},While[!PrimeQ[n(2^k+k)-1],k++];k]; Array[lpi,100] (* Harvey P. Dale, Aug 10 2019 *)

A232109 Least prime p < n + 5 with n + (p-1)*(p-3)/8 prime, or 0 if such a prime p does not exist.

Original entry on oeis.org

5, 3, 3, 5, 3, 5, 3, 7, 11, 5, 3, 5, 3, 7, 17, 5, 3, 5, 3, 7, 11, 5, 3, 23, 17, 7, 11, 5, 3, 5, 3, 13, 11, 7, 19, 5, 3, 7, 17, 5, 3, 5, 3, 7, 17, 5, 3, 23, 11, 7, 11, 5, 3, 23, 17, 7, 11, 5, 3, 5, 3, 31, 11, 7, 19, 5, 3, 7, 11, 5, 3, 5, 3, 13, 17, 7, 19, 5, 3, 7, 17, 5, 3, 23, 17, 7, 11, 5, 3, 29, 11, 13, 11, 7, 19, 5, 3, 7, 11, 5

Offset: 1

Zhi-Wei Sun, Nov 18 2013

nonn

Conjecture: a(n) > 0 for all n > 0. Moreover, for any integer n > 1 there exists a prime p < 2*sqrt(n)*log(7n) such that n + (p-1)*(p-3)/8 is prime.

This implies that any integer n > 1 can be written as (p-1)/2 + q with q a positive integer, and p and (p^2-1)/8 + q both prime.

			a(1) = 5 since neither 1 + (2-1)*(2-3)/8 = 7/8 nor 1 + (3-1)*(3-3)/8 = 1  is prime, but 1 + (5-1)*(5-3)/8 = 2 is prime.

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

Cf. A000040, A185636, A204065, A227898, A228425, A231201, A231557.

Do[Do[If[PrimeQ[n+(Prime[k]-1)(Prime[k]-3)/8],Goto[aa]],{k,1,PrimePi[n+4]}];
Print[n," ",0];Label[aa];Continue,{n,1,100}]

cp's OEIS Frontend

A231633 Number of ways to write n = x + y (x, y > 0) with x^2 * y - 1 prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A231776 Least positive integer k <= n with (2^k + k) * n - 1 prime, or 0 if such a number k does not exist.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A232109 Least prime p < n + 5 with n + (p-1)*(p-3)/8 prime, or 0 if such a prime p does not exist.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica