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A335992 Numbers that are the average of more pairs of distinct twin primes than any previous number. 1
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 (list; graph; refs; listen; history; text; internal format)
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: A278602 A059992 A050570 * A233341 A102110 A302829

Adjacent sequences:  A335989 A335990 A335991 * A335993 A335994 A335995

KEYWORD

nonn

AUTHOR

P. Michael Kielstra, Jul 04 2020

STATUS

approved

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Last modified December 5 11:47 EST 2021. Contains 349557 sequences. (Running on oeis4.)