

A294185


Number of distinct lesser twin primes which are in Goldbach partitions of 2n.


3



0, 0, 1, 2, 2, 1, 2, 3, 2, 2, 4, 3, 1, 3, 2, 2, 5, 3, 0, 4, 3, 2, 5, 5, 1, 4, 3, 1, 5, 3, 2, 6, 3, 0, 6, 5, 2, 6, 6, 0, 6, 5, 1, 6, 5, 1, 4, 3, 0, 7, 5, 2, 5, 6, 2, 9, 7, 1, 8, 6, 0, 6, 4, 0, 8, 5, 1, 3, 7, 2, 9, 7, 0, 7, 5, 2, 9, 6, 0, 9, 5, 0, 7, 11, 1, 6, 6, 1
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OFFSET

1,4


COMMENTS

Tomas Oliveira e Silva in 2012 experimentally confirmed that all even numbers <= 4*10^18 have at least one Goldbach partition (GP) with a prime 9781 or less. Detailed examination of all even numbers < 10^6 showed that the most popular prime in all GPs is 3 (78497 occurrences), then 5 (70328), then 7 (62185), then 11 (48582), then 13 (40916), then 17 (31091), then 19 (29791)  all these primes are twin primes. These results gave rise to a hypothesis that twin primes should be rather frequent in GP, especially those relatively small.
Further empirical experiments demonstrated, surprisingly, there are in general two categories of even numbers n: category 1  with 0, 1, or 2 distinct lesser twin primes in all possible GPs(n), and category 2  with fast increasing number of distinct lesser twin primes in GPs(n).
First occurrence of k, k=0,1,2...: 1, 3, 4, 8, 11, 17, 32, 50, 59, 56, 98, 84, 105, 104, ..., .  Robert G. Wilson v, Jul 24 2018
Records: 0, 1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 14, 15, 17, 20, 22, 25, 28, 32, 33, 36, 37, 43, ..., .  Robert G. Wilson v, Jul 24 2018


LINKS

Robert G. Wilson v, Table of n, a(n) for n = 1..10000
Marcin Barylski, Plot of first 20000 elements of the A294185
Marcin Barylski, C++ program for generating A294185
Tomas Oliveira e Silva, Goldbach conjecture verification


EXAMPLE

a(5) = 2 because 2*5=10 has two ordered Goldbach partitions: 3+7 and 5+5. 3 is a lesser twin prime (because 3 and 5 are twin primes), 5 is a lesser twin prime (because 5 and 7 are twin primes).


MATHEMATICA

a[n_] := Block[{c = 0, p = 3, lst = {}}, While[p < n + 1, If[ PrimeQ[2n  p], AppendTo[lst, {p, 2n  p}]]; p = NextPrime@p]; Length@Select[Union@ Flatten@ lst, PrimeQ[# + 2] &]]; Array[a, 88] (* Robert G. Wilson v, Jul 24 2018 *)


PROG

(C++) See Barylski link.
(PARI) isltwin(p) = isprime(p) && isprime(p+2);
a(n) = {vtp = []; forprime(p = 2, n, if (isprime(2*np), if (isltwin(p), vtp = concat(vtp, p)); if (isltwin(2*np), vtp = concat(vtp, 2*np)); ); ); #Set(vtp); } \\ Michel Marcus, Mar 01 2018


CROSSREFS

Cf. A002372 (number of ordered Goldbach partitions), A001359 (lesser of twin primes), A294186, A295424.
Sequence in context: A261625 A237284 A294186 * A035462 A260414 A160735
Adjacent sequences: A294182 A294183 A294184 * A294186 A294187 A294188


KEYWORD

nonn


AUTHOR

Marcin Barylski, Feb 11 2018


STATUS

approved



