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).

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, 23, 19, 22, 26, 27, 28, 26, 22, 20, 27, 25, 27, 28, 26, 35, 29, 29, 29, 30, 45, 30, 36, 22, 30, 39, 39, 40, 44, 44, 43, 34, 38, 36, 48, 54, 43, 38, 43, 49, 45, 47, 53, 38, 51, 51, 62, 56

Offset: 1

Author

Jaspal Singh Cheema, Feb 04 2010

Keywords

• nonn

Comments

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?

Examples

			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.
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.
		

References

• C. C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Perseus Books, 1999.
• M. D. Sautoy, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, HarperCollins Publishers Inc., 2004.

Links

• J. S. Cheema, Table of n, a(n) for n = 1..1104 (2 prepended by Michael De Vlieger)

Crossrefs

Cf. A023200, A046132, A087679.

Programs

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
nbcp(p) = my(nb=0); forprime(q=p^2, p*(p+4), if (isprime(q+4), nb++)); nb;
lista(nn) = my(v=vcp(nn)); vector(#v, n, nbcp(v[n])); \\ Michel Marcus, Nov 02 2022

Extensions

New name and a(1)=2 prepended by Michel Marcus, Nov 02 2022