A057767 Number of twin prime pairs between P(n)^2 and P(n+1)^2 where P(n) is the n-th prime.

1, 2, 2, 4, 2, 7, 2, 4, 8, 2, 11, 7, 3, 11, 13, 13, 5, 19, 11, 3, 15, 14, 14, 21, 15, 7, 10, 6, 11, 42, 12, 27, 6, 45, 10, 20, 17, 21, 23, 25, 13, 49, 7, 20, 8, 52, 59, 23, 9, 16, 32, 9, 46, 33, 27, 43, 7, 30, 20, 12, 68, 88, 22, 18, 24, 88, 41, 70, 14

Offset: 1

Author

Naohiro Nomoto, Oct 31 2000

Keywords

• nonn

Comments

Conjecture: this sequence is always positive.

For n > 1 also the number of twin ranks k in A002822 between M(n) and M(n+1), where M(n) = (P(n)^2-1)/6. (Indeed, none of the three numbers {6k-1, 6k, 6k+1} will ever be equal to P(n)^2 if 6k+-1 are twin primes, therefore P(n)^2 <= 6k-1 < 6k+1 <= P(n+1)^2 <=> (P(n)^2-1)/6 <= k <= (P(n+1)^2-1)/6.) The twin prime conjecture is equivalent to say a(n) > 0 for infinitely many n. - M. F. Hasler, Jun 26 2019

Records of "lows" (such that a(k) > a(m) for all k > m) are (conjectured): a(1) = 1, a(10) = 2, a(20) = 3, a(33) = 6, a(57) = 7, a(89) = 10, a(140) = 19, a(190) = 21, a(236) = 30, a(256) = 33, a(265) = 35, a(307) = 42, a(346) = 43, a(384) = 44, a(495) = 51, a(498) = 55, a(545) = 62, a(555) = 68, a(613) = 71, a(643) = 76, a(673) = 79, a(719) = 87, a(723) = 93, a(755) = 94, a(772) = 96, a(872) = 98, a(936) = 107, ... None of these is proven, each one would imply the twin prime conjecture. - M. F. Hasler, Jun 26 2019

Record lows of a(n)/n are: 1/1 = 2/2 = 1.0, 2/3 = 0.66667, 2/5 = 0.4, 2/7 = 0.28571, 2/10 = 0.2, 3/20 = 0.15, 7/57 = 0.12281, 10/89 = 0.11236, 21/190 = 0.11053, 51/495 = 0.10303, 342/3435 = 0.099563, 716/7202 = 0.099417, 797/8126 = 0.098080, 793/8155 = 0.097241, 817/8463 = 0.096538, 892/9406 = 0.094833, ... - M. F. Hasler, Jun 27 2019

Examples

			From _M. F. Hasler_, Jun 26 2019: (Start)
Between P(1)^2 = 2^2 = 4 and P(2)^2 = 3^2 = 9 there is only the twin prime pair (5,7), whence a(1) = 1.
Between P(2)^2 = 3^2 = 9 and P(3)^2 = 5^2 = 25 there are the twin prime pairs (11,13) and (17,19) whence a(2) = 2.
Between P(3)^2 = 5^2 = 25 and P(4)^2 = 7^2 = 49 there are the twin prime pairs (29,31) and (41,43) whence a(3) = 2.
Between P(4)^2 = 7^2 = 49 and P(5)^2 = 11^2 = 121 there are the twin prime pairs (59,61), (71,73), (101,103) and (107,109), whence a(4) = 4.
etc. (End)
		

Links

• M. F. Hasler, Table of n, a(n) for n = 1..10000
• A. Dinculescu, On the Numbers that Determine the Distribution of Twin Primes, Surveys in Mathematics and its Applications, 13 (2018), 171-181.

Programs

Mathematica

cp[{x_,y_}]:=Count[Partition[Range[x+1,y-1],3,1],?(AllTrue[{#[[1]],#[[3]]},PrimeQ]&)]; cp/@ Partition[Prime[Range[100]]^2,2,1] (* _Harvey P. Dale, Jul 27 2024 *)

PARI

A057767(n,c=0)={forprime(q=2+p=nextprime(prime(n)^2),prime(n+1)^2,p+2==(p=q)&&c++); c} \\ (Replaces slower code from Jun 26 2019.) - M. F. Hasler, Jul 04 2019

Extensions

Offset corrected to 1 by M. F. Hasler, Jun 26 2019