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A172483 a(n) is the number of cousin primes between p^2 and p*(p+4) where p is the n-th cousin prime A023200(n).

%I A172483 #16 Nov 03 2022 10:08:54
%S A172483 2,1,1,2,5,4,4,2,6,4,7,7,5,9,12,13,14,14,9,12,10,11,13,20,16,15,16,15,
%T A172483 23,19,22,26,27,28,26,22,20,27,25,27,28,26,35,29,29,29,30,45,30,36,22,
%U A172483 30,39,39,40,44,44,43,34,38,36,48,54,43,38,43,49,45,47,53,38,51,51,62,56
%N A172483 a(n) is the number of cousin primes between p^2 and p*(p+4) where p is the n-th cousin prime A023200(n).
%C A172483 If you graph the order of the consecutive cousin primes along the x-axis (i.e., first pair of cousin primes, second, third,...) and the number of cousin primes in the sequence given above along the y-axis, a clear pattern emerges. As you go farther along the x-axis, greater are the number of consecutive cousin primes, on average, within the interval obtained. If one can prove that there's at least one consecutive cousin prime within each interval, this would imply that cousin primes are infinite. I suspect the number of consecutive primes within each interval will never be zero. Can you prove it?
%D A172483 C. C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Perseus Books, 1999.
%D A172483 M. D. Sautoy, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, HarperCollins Publishers Inc., 2004.
%H A172483 J. S. Cheema, <a href="/A172483/b172483.txt">Table of n, a(n) for n = 1..1104</a> (2 prepended by Michael De Vlieger)
%e A172483 The 1st pair of cousin primes is (3, 7), between 3^2=9 and 3*7=21 there is 2 cousin primes: 13 and 19. So a(1) = 2.
%e A172483 The 2nd pair of cousin primes is (7, 11), between 7^2=49 and 7*11=77 there is 1 cousin prime: 67. So a(2) = 1.
%o A172483 (PARI) vcp(nn) = my(list=List(), p=3); listput(list, p); p=7; forprime(q=11, nn, if(q-p==4, listput(list, p)); p=q); Vec(list); \\ A023200
%o A172483 nbcp(p) = my(nb=0); forprime(q=p^2, p*(p+4), if (isprime(q+4), nb++)); nb;
%o A172483 lista(nn) = my(v=vcp(nn)); vector(#v, n, nbcp(v[n])); \\ _Michel Marcus_, Nov 02 2022
%Y A172483 Cf. A023200, A046132, A087679.
%K A172483 nonn
%O A172483 1,1
%A A172483 _Jaspal Singh Cheema_, Feb 04 2010
%E A172483 New name and a(1)=2 prepended by _Michel Marcus_, Nov 02 2022