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Number of twin prime pairs between P(n)^2 and P(n+1)^2 where P(n) is the n-th prime.
5

%I #27 Jul 27 2024 17:17:59

%S 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,

%T 11,42,12,27,6,45,10,20,17,21,23,25,13,49,7,20,8,52,59,23,9,16,32,9,

%U 46,33,27,43,7,30,20,12,68,88,22,18,24,88,41,70,14

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

%C Conjecture: this sequence is always positive.

%C 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

%C 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

%C 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

%H M. F. Hasler, <a href="/A057767/b057767.txt">Table of n, a(n) for n = 1..10000</a>

%H A. Dinculescu, <a href="http://www.utgjiu.ro/math/sma/v13/p13_11.pdf">On the Numbers that Determine the Distribution of Twin Primes</a>, Surveys in Mathematics and its Applications, 13 (2018), 171-181.

%e From _M. F. Hasler_, Jun 26 2019:

%e 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.

%e 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.

%e 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.

%e 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.

%e etc. (End)

%t 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 *)

%o (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

%K nonn

%O 1,2

%A _Naohiro Nomoto_, Oct 31 2000

%E Offset corrected to 1 by _M. F. Hasler_, Jun 26 2019