This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A341843 #88 Feb 16 2025 08:34:01
%S A341843 0,0,0,0,1,4,7,13,25,45,80,136,251,443,784,1377,2420,4312,7756,14106,
%T A341843 25554,46776,85774,157325,290773,538520,1000321,1861364,3473165,
%U A341843 6493997,12167342,22851920,42987462,81018661,152945700,289206487,547722346,1038786862
%N A341843 Number of sexy consecutive prime pairs below 2^n.
%C A341843 a(n) is the number of pairs of consecutive sexy primes {A023201, A046117} less than 2^n.
%C A341843 For each n from 9 through 48, the most frequently occurring difference between consecutive primes is 6. On p. 108 of the article by Odlyzko et al., the authors estimate that around n=117, the jumping champion (i.e., the most frequently occurring difference between consecutive primes) becomes 30, and around n=1412 it becomes 210. Successive jumping champions are conjecturaly the primorial numbers A002110.
%C A341843 Data for n >= 15 taken from Marek Wolf's prime gaps computation.
%C A341843 For the number of pairs of consecutive primes below 10^n having a difference of 6, see A093738.
%C A341843 For the number of sexy primes less than 10^n, see A080841.
%C A341843 There are 8 known cases in which a power of 2 falls between the members of the sexy consecutive prime pair (see A220951), but if a pair (p, p+6) is such that p < 2^n < p+6, that pair is not counted in a(n).
%H A341843 Artur Jasinski, <a href="/A341843/b341843.txt">Table of n, a(n) for n = 1..48</a>
%H A341843 Andrew Odlyzko, Michael Rubinstein, Marek Wolf, <a href="https://projecteuclid.org/journals/experimental-mathematics/volume-8/issue-2/Jumping-champions/em/1047477055.full">Jumping-champions</a>, Experimental Mathematics 8:2, pp. 108-118, 1999.
%H A341843 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 A341843 Marek Wolf, <a href="http://pracownicy.uksw.edu.pl/mwolf/gaps.zip">Counted prime gaps for range x from 2^15 to 2^48</a>.
%e A341843 a(6)=4 because 2^6=64 and we have 4 sexy consecutive prime pairs less than 64: {23,29}, {31,37}, {47,53}, {53,59}.
%t A341843 pp = {}; Do[kk = 0; Do[If[Prime[m + 1] - Prime[m] == 6, kk = kk + 1], {m, 2, PrimePi[2^n] - 1}]; AppendTo[pp, kk], {n, 4, 20}]; pp
%Y A341843 Cf. A002110, A023201, A046117, A080841, A098428, A104037, A220951, A226068, A227346.
%K A341843 nonn,hard
%O A341843 1,6
%A A341843 _Artur Jasinski_, Feb 21 2021