cp's OEIS Frontend

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

%I A173198 #12 Feb 16 2025 08:33:11
%S A173198 10,10,12,8,11,14,12,15,18,19,21,21,25,31,19,23,32,29,27,28,43,36,36,
%T A173198 35,42,51,52,46,43,53,45,55,41,55,51,46,71,52,66,60,54,62,75,66,56,67,
%U A173198 91,65,78,75,77,97,62,80,90,81,68,78,89,99,86,90,98,98,106,96,90,84,105,89
%N A173198 Number of pairs of sexy consecutive primes between (A031924(n))^2 and A031924(n)*A031925(n).
%C A173198 If you graph a(n) versus n, a clear pattern emerges.
%C A173198 As you go farther along the n-axis, greater are the number of consecutive sexy primes, on average, within each interval obtained.
%C A173198 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.
%C A173198 I suspect all numbers in the sequence are > 0.
%H A173198 J. S. Cheema, <a href="/A173198/b173198.txt">Table of n, a(n) for n = 1..1762</a>
%H A173198 Rick Aster, <a href="http://www.globalstatements.com/shortcuts/88a.html">Prime number sieve</a> SAS prime sieve program
%H A173198 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SexyPrimes.html">Sexy Primes</a>. [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].
%H A173198 Wikipedia, <a href="http://en.wikipedia.org/wiki/Sexy_primes">Sexy Primes</a>
%e A173198 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.
%e A173198 Between these two numbers, namely (529,667), there are ten consecutive sexy primes: (541,547), (557,563), (563,569),
%e A173198 (571,577), (587,593), (593,599), (601,607), (607,613), (647,653), and (653 659).
%e A173198 Hence the very first term of the sequence is 10.
%p A173198 isA031924 := proc(p) return (isprime(p) and (nextprime(p)-p) = 6 ); end proc:
%p A173198 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:
%p A173198 A031925 := proc(n) A031924(n)+6 ; end proc:
%p A173198 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:
%p A173198 seq(A173198(n),n=1..80) ; # _R. J. Mathar_, Feb 15 2010
%Y A173198 Cf. A023201
%K A173198 nonn
%O A173198 1,1
%A A173198 _Jaspal Singh Cheema_, Feb 12 2010
%E A173198 Comments condensed by _R. J. Mathar_, Feb 15 2010