A090695 -id:A090695 - cp's OEIS frontend

A091182 Number of ways to write n = x + y (x >= y > 0) with xy - 1 and xy + 1 both prime.

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

Offset: 1

Ray Chandler, Dec 27 2003

From Zhi-Wei Sun, Nov 27 2012: (Start)
Conjecture: a(n) > 0 for all n > 3120. This has been verified for n up to 5*10^7.
Note that if x >= y > 0 and x+y = n then n-1 = x+y-1 <= xy <= ((x+y)/2)^2 = n^2/4. So the conjecture implies that there are infinitely many twin primes.
For n=4,5,...,3120 we can write n = x+y (x >= y > 0) with xy-1 prime.
For each positive integer n <= 3120 different from 1,6,30,54, we can write n = x+y (x >= y > 0) with xy+1 prime.
More generally, we have the following conjecture: Let m be any positive integer. If n is sufficiently large and (m-1)n is even, then we can write n as x+y, where x and y are positive integers with xy-m and xy+m both prime. This general conjecture implies that for any positive even integer d there are infinitely many primes p and q with difference d. (End)
Sequence A090695 lists the 61 known values of n where a(n) = 0. - T. D. Noe, Nov 29 2012

			a(8)=1 since 8=6+2 with 6*2-1 and 6*2+1 both prime.
a(11)=2 since 11=6+5=9+2 with 6*5-1, 6*5+1, 9*2-1, 9*2+1 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..50000 (first 10000 terms from T. D. Noe)
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A001359, A006512, A014574, A090695, A091183, A219157.

with(numtheory); a:=n->sum( (pi((i)*(n-i)+1) - pi((i)*(n-i)))*(pi((i)*(n-i)-1) - pi((i)*(n-i) - 2)) , i=1..floor(n/2) ); seq(a(k),k=1..100); # Wesley Ivan Hurt, Jan 21 2013

Table[cnt = 0; Do[If[PrimeQ[k*(n - k) - 1] && PrimeQ[k*(n - k) + 1], cnt++], {k, n/2}]; cnt, {n, 100}] (* Zhi-Wei Sun, edited by T. D. Noe, Nov 29 2012 *)

Edited by N. J. A. Sloane, Nov 29 2012

A091183 Index of first occurrence of n in A091182.

Original entry on oeis.org

1, 4, 5, 13, 17, 71, 31, 53, 109, 67, 97, 143, 197, 101, 253, 223, 137, 193, 277, 319, 461, 269, 547, 367, 473, 353, 283, 529, 361, 563, 889, 431, 577, 809, 661, 739, 857, 599, 913, 1081, 899, 973, 811, 1109, 613, 647, 1061, 781, 1369, 907, 1373, 1039, 1381

Offset: 0

Ray Chandler, Dec 27 2003

nonn

Cf. A014574, A090695, A091182.

A091417 Integers n which are not the difference of 2 integers A and B where A is less than or equal to n such that AB+1 and AB-1 are both primes. In other words, the product A*B cannot be the middle integer of a twin prime pair.

Original entry on oeis.org

8, 9, 10, 15, 18, 20, 30, 40, 51, 58, 60, 65, 75, 96, 100, 110, 120, 170, 195, 240, 250, 260, 315, 330, 345, 390, 540, 630, 750, 780, 910, 1020, 1720, 1890

Offset: 1

William F. Sindelar (w_sindelar(AT)juno.com), Jan 05 2004

Sequence may be finite. Next term after 1890 if it exists must be greater than 867750.

			8 is a member: 8 is the difference between these pairs of integers: (9-1) (10-2) (11-3) (12-4) (13-5) (14-6) (15-7) (16-8). Their products (9*1) (10*2), etc. plus and minus 1 are not primes and therefore the products cannot be the middle integers of twin prime sets.

Cf. A090695, A091182, A091183. A091418.

A091418 Integers n which are not the sum or difference of 2 integers A and B where A is less than or equal to n such that AB+1 and AB-1are both primes. In other words, the product A*B cannot be the middle integer of a twin prime pair.

Original entry on oeis.org

10, 15, 20, 30, 40, 60, 75, 100, 120, 250, 315, 390, 540, 630

Offset: 1

William F. Sindelar (w_sindelar(AT)juno.com), Jan 05 2004

Sequence may be finite. Next term after 630 if it exists must be greater than 867750.

The terms in this sequence are those which are common to A090695 and A091417.

			See A090695 and A091417 for examples.

Cf. A090695, A091417, A091182, A091183.

cp's OEIS Frontend

A091182 Number of ways to write n = x + y (x >= y > 0) with xy - 1 and xy + 1 both prime.

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A091183 Index of first occurrence of n in A091182.

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A091417 Integers n which are not the difference of 2 integers A and B where A is less than or equal to n such that AB+1 and AB-1 are both primes. In other words, the product A*B cannot be the middle integer of a twin prime pair.

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A091418 Integers n which are not the sum or difference of 2 integers A and B where A is less than or equal to n such that AB+1 and AB-1are both primes. In other words, the product A*B cannot be the middle integer of a twin prime pair.

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