login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A257474 Number of unordered ways to write n = a + b, where a and b are distinct elements of the set {floor(x/3): 3*x-1 and 3*x+1 are twin prime}, one of a and b is even, and one of a and b has the form p-1 or p-2 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 (list; graph; refs; listen; history; text; internal format)
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

Zhi-Wei Sun, Table of n, a(n) for n = 1..20000

Zhi-Wei 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*76-1,3*76+1} = {227,229} and {3*540-1,3*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*128-1,3*128+1} = {1151,1153} and {3*540-1,3*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*930-1,3*930+1} ={2789,2791} and {3*934-1,3*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*934-1,3*934+1} = {2801,2803} and {3*1274-1,3*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*46-1,3*46+1} = {137,139} and {3*13920-1,3*13920+1} = {41759,41761} twin prime pairs.

MATHEMATICA

TQ[n_]:=PrimeQ[3n-1]&&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(n-x), 2]==0&&(RQ[x]||RQ[n-x])&&PQ[x]&&PQ[n-x], r=r+1], {x, 0, (n-1)/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

Zhi-Wei Sun, Apr 25 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 26 11:07 EST 2020. Contains 338639 sequences. (Running on oeis4.)