A335992 Numbers that are the average of more pairs of distinct twin primes than any previous number.

1, 4, 8, 12, 24, 57, 105, 150, 330, 645, 666, 945, 1155, 1770, 1785, 2625, 2925, 3255, 3465, 5145, 5460, 5775, 6930, 8295, 10605, 11340, 13650, 15015, 17205, 18480, 19635, 21945, 27930, 30030, 38115, 42735, 45045, 48840, 51765, 53130

Offset

1,2

Comments

Let T(n) be the number of pairs of twin primes (that is, primes p where p+2 or p-2 is also prime) with average n. These are the positions at which T(n) attains high-water marks.

Links

Table of n, a(n) for n=1..40.

N. J. A. Sloane, Transforms (The RECORDS transform returns both the high-water marks and the places where they occur).

Example

1 is not the average of any pairs of twin primes.  4 is the average of one pair of twin primes: 3 and 5.  8 is the average of two pairs of twin primes: 5 and 11, and 3 and 13.  (Note that the difference between the twin primes in each pair is not necessarily 2.  However, both members of the pair are twin primes, that is, prime numbers p such that either p+2 or p-2 is also prime.  The fact that their twins are not part of the pair doesn't matter.)

Mathematica

m = 10^4; tp = Select[Range[3, m, 2], PrimeQ[#] && Or @@ PrimeQ[# + {-2, 2}] &]; f[n_] := Module[{k = Length @ IntegerPartitions[n, {2}, tp]}, If[MemberQ[tp, n/2], k - 1, k]]; s = {}; fm = 0; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 2, m/2, 2}]; Prepend[s/2, 1] (* Amiram Eldar, Jul 11 2020 *)

Crossrefs

The values attained at these high-water marks are given in A335993.

Cf. A001097, A053033.

Sequence in context: A382327 A059992 A050570 * A394918 A233341 A102110

Adjacent sequences: A335989 A335990 A335991 * A335993 A335994 A335995

Keyword

nonn

Author

P. Michael Kielstra, Jul 04 2020

Status

approved