A256707 -id:A256707 - cp's OEIS frontend

A257121 Numbers m with 9m + 3r - 1 and 9m + 3r + 1 twin prime for some r = 0,1,2.

0, 1, 2, 3, 4, 6, 8, 11, 12, 15, 16, 20, 21, 22, 25, 26, 30, 31, 34, 38, 46, 48, 51, 58, 63, 66, 68, 71, 73, 90, 91, 92, 95, 98, 113, 114, 116, 118, 121, 128, 136, 142, 143, 144, 146, 158, 161, 164, 165, 178, 180, 185, 188, 191, 198, 208, 214, 216, 222, 225, 231, 232, 234, 236, 238, 248, 252, 256, 260, 264, 283, 288, 295, 298, 301, 303, 310, 311, 330, 333

Offset: 1

Zhi-Wei Sun, Apr 25 2015

nonn

By the conjecture in A257317, any positive integer should be the sum of two distinct terms of the current sequence one of which is even.

			a(1) = 0 since {9*0+3*2-1,9*0+3*2+1} = {5,7} is a twin prime pair.
a(2) = 1 since {9*1+3*1-1,9*1+3*1+1} = {11,13} is a twin prime pair.
a(3) = 2 since {9*2+3*0-1,9*2+3*0+1} = {17,19} is a twin prime pair.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
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.

Cf. A014574, A256707, A257317.

TQ[m_]:=PrimeQ[3m-1]&&PrimeQ[3m+1]
PQ[m_]:=TQ[3*m]||TQ[3*m+1]||TQ[3*m+2]
n=0;Do[If[PQ[m],n=n+1;Print[n," ",m]],{m,0,340}]

A257317 Number of unordered ways to write n as the sum of two distinct elements of the set {floor(x/3): 3x-1 and 3x+1 are twin prime} one of which is even.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Apr 25 2015

nonn

Conjecture: a(n) > 0 for all n > 0.

Clearly, this conjecture implies the Twin Prime Conjecture. Note that a(n) does not exceed A256707(n).

			a(4) = 1 since 4 = 0 + 4 = floor(2/3) + floor(14/3) with 0 or 4 even, and {3*2-1,3*2+1} = {5,7} and {3*14-1,3*14+1} = {41,43} twin prime pairs.
a(108) = 1 since 108 = 16 + 92 = floor(50/3) + floor(276/3) with 16 or 92 even, and {3*50-1,3*50+1} = {149,151} and {3*276-1,3*276+1} = {827,829} twin prime pairs.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
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.

Cf. A014574, A256707, A257121.

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

A257474 Number of unordered ways to write n = a + b, where a and b are distinct elements of the set {floor(x/3): 3x-1 and 3x+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.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Apr 25 2015

nonn

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.

			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.

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.

Cf. A000040, A014574, A256707, A257121, A257317.

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}]

A257497 Number of ordered ways to write n as the sum of a term of A257121 and a positive generalized pentagonal number.

Original entry on oeis.org

1, 2, 2, 2, 3, 2, 3, 3, 3, 2, 2, 2, 5, 2, 3, 4, 4, 4, 2, 2, 3, 4, 6, 3, 2, 5, 7, 5, 2, 4, 3, 5, 4, 3, 4, 4, 6, 5, 3, 3, 5, 4, 5, 2, 2, 5, 4, 4, 2, 3, 5, 5, 6, 1, 4, 5, 4, 3, 3, 7, 4, 2, 5, 2, 5, 4, 2, 4, 3, 6, 4, 5, 9, 4, 3, 3, 4, 8, 2, 4, 5, 3, 5, 1, 5, 4, 1, 5, 3, 2, 4, 6, 6, 3, 5, 4, 6, 5, 5, 5

Offset: 1

Zhi-Wei Sun, Apr 26 2015

nonn

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 54, 84, 87, 109, 174, 252, 344, 1234, 1439, 2924.

This implies the Twin Prime Conjecture.

			a(1439) = 1 since 1439 = 1424 + 15 = floor(4274/3) + (-3)*(3*(-3)-1)/2 with {3*4274-1,3*4274+1} = {12821,12823} a twin prime pair.
a(2924) = 1 since 2924 = 2334 + 590 = floor(7004/3) + 20*(3*20-1)/2 with {3*7004-1, 3*7004+1} = {21011,21013} a twin prime pair.

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.

Cf. A001318, A014574, A257121, A256071, A256707, A257317, A257474.

TQ[n_]:=PrimeQ[3n-1]&&PrimeQ[3n+1]
PQ[n_]:=TQ[3*n]||TQ[3*n+1]||TQ[3n+2]
SQ[n_]:=IntegerQ[Sqrt[24n+1]]
Do[m=0;Do[If[PQ[x]&&SQ[n-x],m=m+1],{x,0,n-1}];
Print[n," ",m];Continue,{n,1,100}]

cp's OEIS Frontend

A257121 Numbers m with 9*m + 3*r - 1 and 9*m + 3*r + 1 twin prime for some r = 0,1,2.

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A257317 Number of unordered ways to write n as the sum of two distinct elements of the set {floor(x/3): 3*x-1 and 3*x+1 are twin prime} one of which is even.

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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.

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Programs

Mathematica

A257497 Number of ordered ways to write n as the sum of a term of A257121 and a positive generalized pentagonal number.

Views

Author

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Examples

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Crossrefs

Programs

Mathematica

A257121 Numbers m with 9m + 3r - 1 and 9m + 3r + 1 twin prime for some r = 0,1,2.

A257317 Number of unordered ways to write n as the sum of two distinct elements of the set {floor(x/3): 3x-1 and 3x+1 are twin prime} one of which is even.

A257474 Number of unordered ways to write n = a + b, where a and b are distinct elements of the set {floor(x/3): 3x-1 and 3x+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.