A174222 Number of symmetric primes in the interval [prime(n)^2, prime(n)*prime(n+1)].

1, 2, 2, 6, 4, 7, 5, 10, 18, 6, 24, 18, 10, 21, 35, 29, 14, 33, 27, 14, 44, 32, 43, 64, 36, 16, 36, 17, 38, 133, 41, 71, 16, 123, 21, 71, 72, 49, 90, 85, 36, 158, 34, 66, 31, 190, 184, 73, 39, 73, 109, 33, 188, 109, 117, 110, 35, 126, 85, 36, 221, 298, 99, 41, 95, 320, 136, 237

Offset

1,2

Comments

If you graph a(n) vs. n, an interesting pattern with random-looking fluctuations begins to emerge.

As you go farther along the n-axis, greater are the number of symmetric primes, on average.

The smallest count of a(.)=1 occurs only once at the very beginning.

I suspect all a(n) are > 0. If one could prove this, it would imply that Symmetric primes are infinite.

Links

Table of n, a(n) for n=1..68.

Formula

#{ A090190(j): A001248(n) < A090190(j) < A006094(n)}.

Example

The square of the first prime is 2^2=4 and the product of the first and second prime is 2*3=6. Within this interval, there is 1 symmetric prime, which is 5. Hence a(1)=1.
The second term, a(2)=2, refers to the two symmetric primes 11 and 13 within the interval (9,15).

Prog

(PARI) issym(p) = fordiv(p-1, d, if(isprime(p-d) || isprime(p+d), return(1))); 0; \\ A090190
a(n) = my(p=prime(n), nb=0); forprime(q=p^2, p*nextprime(p+1), if (issym(q), nb++)); nb; \\ Michel Marcus, Nov 03 2022

Crossrefs

Cf. A090190, A090191.

Cf. A006094, A001248,

Sequence in context: A182411 A067804 A074911 * A071059 A256468 A061108

Adjacent sequences: A174219 A174220 A174221 * A174223 A174224 A174225

Keyword

nonn

Author

Jaspal Singh Cheema, Mar 18 2010

Extensions

Edited by R. J. Mathar, Mar 31 2010

Status

approved