A341843 Number of sexy consecutive prime pairs below 2^n.

0, 0, 0, 0, 1, 4, 7, 13, 25, 45, 80, 136, 251, 443, 784, 1377, 2420, 4312, 7756, 14106, 25554, 46776, 85774, 157325, 290773, 538520, 1000321, 1861364, 3473165, 6493997, 12167342, 22851920, 42987462, 81018661, 152945700, 289206487, 547722346, 1038786862

Offset: 1

Author

Artur Jasinski, Feb 21 2021

Keywords

• nonn
• hard

Comments

a(n) is the number of pairs of consecutive sexy primes {A023201, A046117} less than 2^n.

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.

Data for n >= 15 taken from Marek Wolf's prime gaps computation.

For the number of pairs of consecutive primes below 10^n having a difference of 6, see A093738.

For the number of sexy primes less than 10^n, see A080841.

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).

Examples

			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}.
		

Links

• Artur Jasinski, Table of n, a(n) for n = 1..48
• Andrew Odlyzko, Michael Rubinstein, Marek Wolf, Jumping-champions, Experimental Mathematics 8:2, pp. 108-118, 1999.
• 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].
• Marek Wolf, Counted prime gaps for range x from 2^15 to 2^48.

Crossrefs

Cf. A002110, A023201, A046117, A080841, A098428, A104037, A220951, A226068, A227346.

Programs

Mathematica

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