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

Author

P. Michael Kielstra, Jul 04 2020

Keywords

• nonn

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.

Examples

			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.)
		

Links

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

Crossrefs

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

Cf. A001097, A053033.

Programs

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 *)