

A256707


Number of unordered ways to write n as the sum of two distinct elements of the set {floor(x/3): 3*x1 and 3*x+1 are twin prime}.


5



1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 5, 4, 3, 5, 5, 5, 3, 3, 5, 5, 5, 5, 3, 5, 6, 5, 2, 3, 5, 6, 2, 1, 3, 7, 4, 3, 4, 5, 5, 5, 3, 5, 3, 4, 3, 3, 5, 4, 3, 3, 4, 5, 2, 5, 4, 5, 6, 4, 5, 6, 5, 7, 3, 4, 5, 4, 6, 3, 3, 4, 4, 5, 3, 3, 2, 5, 5, 2, 4, 6, 7, 7, 4, 6, 6, 6, 6, 3
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OFFSET

1,3


COMMENTS

Conjecture: a(n) > 0 for all n > 0. Moreover, for any integer m > 10 every positive integer can be written as the sum of two distinct elements of the set {floor(x/m): x1 and x+1 are twin prime}.
Clearly, the conjecture implies the Twin Prime Conjecture.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
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(44) = 1 since 44 = 6 + 38 = floor(20/3) + floor(116/3) with {3*201,3*20+1} = {59,61} and {3*1161,3*116+1} = {347,349} twin prime pairs.
a(108) = 1 since 108 = 16 + 92 = floor(50/3) + floor(276/3) with {3*501,3*50+1} = {149,151} and {3*2761,3*276+1} = {827,829} twin prime pairs.


MATHEMATICA

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


CROSSREFS

Cf. A014574, A256555.
Sequence in context: A074908 A307713 A328680 * A151963 A191291 A131841
Adjacent sequences: A256704 A256705 A256706 * A256708 A256709 A256710


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Apr 24 2015


STATUS

approved



