

A257474


Number of unordered ways to write n = a + b, where a and b are distinct elements of the set {floor(x/3): 3*x1 and 3*x+1 are twin prime}, one of a and b is even, and one of a and b has the form p1 or p2 with p prime.


2



1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 3, 5, 3, 3, 3, 5, 3, 3, 3, 5, 3, 4, 3, 3, 3, 6, 5, 1, 2, 5, 4, 2, 1, 2, 3, 4, 3, 4, 5, 3, 3, 3, 3, 3, 2, 2, 2, 4, 3, 3, 2, 3, 3, 1, 3, 4, 4, 5, 4, 4, 3, 4, 3, 3, 1, 5, 3, 5, 3, 2, 1, 3, 3, 3, 1, 2, 2, 4, 2, 4, 4, 5, 3, 4, 4, 5, 3, 3, 2
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OFFSET

1,3


COMMENTS

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 2, 4, 39, 44, 65, 76, 82, 86, 108, 110, 123, 154, 175, 178, 196, 205, 221, 284, 308, 621, 735, 4655.
This is much stronger than the Twin Prime Conjecture. Note that a(n) <= A257317(n) <= A256707(n). We have verified that a(n) > 0 for all n = 1..10^7.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..20000
ZhiWei Sun, Natural numbers represented by floor(x^2/a)+floor(y^2/b)+floor(z^2/c), arXiv:1504.01608 [math.NT], 2015.


EXAMPLE

a(205) = 1 since 205 = 25 + 180 = floor(76/3) + floor(540/3) with 180 even and 180 + 1 prime, and {3*761,3*76+1} = {227,229} and {3*5401,3*540+1} = {1619,1621} twin prime pairs.
a(308) = 1 since 308 = 128 + 180 = floor(384/3) + floor(540/3) with 180 + 1 prime, and {3*1281,3*128+1} = {1151,1153} and {3*5401,3*540+1} = {1619,1621} twin prime pairs.
a(621) = 1 since 621 = 310 + 311 = floor(930/3) + floor(934/3) with 310 even and 310 + 1 prime, {3*9301,3*930+1} ={2789,2791} and {3*9341,3*934+1} = {2801,2803} twin prime pairs.
a(735) = 1 since 735 = 311 + 424 = floor(934/3) + floor(1274/3) with 424 even, 311 + 2 = 313 prime, and {3*9341,3*934+1} = {2801,2803} and {3*12741,3*1274+1} = {3821,3823} twin prime pairs.
a(4655) = 1 since 4655 = 15 + 4640 = floor(46/3) + floor(13920/3) with 4640 even, 15 + 2 prime, and {3*461,3*46+1} = {137,139} and {3*139201,3*13920+1} = {41759,41761} twin prime pairs.


MATHEMATICA

TQ[n_]:=PrimeQ[3n1]&&PrimeQ[3n+1]
PQ[n_]:=TQ[3*n]TQ[3*n+1]TQ[3n+2]
RQ[n_]:=PrimeQ[n+1]PrimeQ[n+2]
Do[r=0; Do[If[Mod[x(nx), 2]==0&&(RQ[x]RQ[nx])&&PQ[x]&&PQ[nx], r=r+1], {x, 0, (n1)/2}];
Print[n, " ", r]; Continue, {n, 1, 100}]


CROSSREFS

Cf. A000040, A014574, A256707, A257121, A257317.
Sequence in context: A237684 A130634 A274828 * A257317 A163376 A261913
Adjacent sequences: A257471 A257472 A257473 * A257475 A257476 A257477


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Apr 25 2015


STATUS

approved



