A173198 Number of pairs of sexy consecutive primes between (A031924(n))^2 and A031924(n)*A031925(n).

10, 10, 12, 8, 11, 14, 12, 15, 18, 19, 21, 21, 25, 31, 19, 23, 32, 29, 27, 28, 43, 36, 36, 35, 42, 51, 52, 46, 43, 53, 45, 55, 41, 55, 51, 46, 71, 52, 66, 60, 54, 62, 75, 66, 56, 67, 91, 65, 78, 75, 77, 97, 62, 80, 90, 81, 68, 78, 89, 99, 86, 90, 98, 98, 106, 96, 90, 84, 105, 89

Offset: 1

Author

Jaspal Singh Cheema, Feb 12 2010

Keywords

• nonn

Comments

If you graph a(n) versus n, a clear pattern emerges.

As you go farther along the n-axis, greater are the number of consecutive sexy primes, on average, within each interval obtained.

If one could prove that there is at least one consecutive sexy prime within each interval, this would imply that consecutive sexy primes are infinite.

I suspect all numbers in the sequence are > 0.

Examples

			The first sexy prime pair with consecutive primes is (23,29) = A031924(1) and A031925(1). Square the first term, you get 529, then take the product of the two primes, you get 667.
Between these two numbers, namely (529,667), there are ten consecutive sexy primes: (541,547), (557,563), (563,569),
(571,577), (587,593), (593,599), (601,607), (607,613), (647,653), and (653 659).
Hence the very first term of the sequence is 10.
		

Links

• J. S. Cheema, Table of n, a(n) for n = 1..1762
• Rick Aster, Prime number sieve SAS prime sieve program
• Eric Weisstein's World of Mathematics, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes.- _N. J. A. Sloane_, Mar 07 2021].
• Wikipedia, Sexy Primes

Crossrefs

Cf. A023201

Programs

Maple

isA031924 := proc(p) return (isprime(p) and (nextprime(p)-p) = 6 ); end proc:
A031924 := proc(n) local p; if n = 1 then 23; else p := nextprime(procname(n-1)) ; while not isA031924(p) do p := nextprime(p) ; end do ; return p ; end if ; end proc:
A031925 := proc(n) A031924(n)+6 ; end proc:
A173198 := proc(n) local ulim,llim,a,i ; llim := A031924(n)^2 ; ulim := A031924(n)*A031925(n) ; a := 0 ; for i from llim to ulim-6 do if isA031924(i) then a := a+1 ; end if; end do ; a ; end proc:
seq(A173198(n),n=1..80) ; # R. J. Mathar, Feb 15 2010

Extensions

Comments condensed by R. J. Mathar, Feb 15 2010