|
%I #19 Aug 03 2020 15:35:47
%S 1,4,8,12,24,57,105,150,330,645,666,945,1155,1770,1785,2625,2925,3255,
%T 3465,5145,5460,5775,6930,8295,10605,11340,13650,15015,17205,18480,
%U 19635,21945,27930,30030,38115,42735,45045,48840,51765,53130
%N Numbers that are the average of more pairs of distinct twin primes than any previous number.
%C 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.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a> (The RECORDS transform returns both the high-water marks and the places where they occur).
%e 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.)
%t 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 *)
%Y The values attained at these high-water marks are given in A335993.
%Y Cf. A001097, A053033.
%K nonn
%O 1,2
%A _P. Michael Kielstra_, Jul 04 2020
|
