A242758 -id:A242758 - cp's OEIS frontend

A242767 Numbers of repetitions of terms in A242758.

1, 1, 2, 2, 3, 3, 4, 3, 6, 2, 5, 2, 6, 2, 2, 4, 3, 5, 3, 4, 5, 12, 2, 6, 9, 6, 5, 4, 3, 4, 20, 2, 2, 4, 4, 19, 2, 3, 2, 4, 8, 11, 5, 3, 3, 3, 10, 5, 4, 2, 17, 3, 6, 3, 3, 9, 9, 2, 6, 2, 6, 5, 6, 2, 3, 2, 3, 9, 4, 7, 3, 7, 20, 4, 7, 6, 5, 3, 7, 3, 20, 2, 14, 4

Offset: 2

Vladimir Shevelev, May 22 2014

nonn

If {pA242758. If this number occurs k times in A242758, then we say that k is the index of the pair of twin primes {p,q} with p in A001359.

Is this the same as A027833 shifted by two indices? - R. J. Mathar, May 23 2014

Peter J. C. Moses, Table of n, a(n) for n = 2..5001

Cf. A001359, A006512, A242489, A242490, A242720, A242758.

From the construction of A242758, in supposition of an infinity of twin primes, we have a(2)=1; for n>=3, a(n) = A027833(n-2). Otherwise, A027833 is finite, while A242758 will coincide with A242720 after the last pair of twin primes. - Vladimir Shevelev, May 26 2014

More terms from Peter J. C. Moses, May 22 2014

A242768 Indices of twin primes in {A242758(n)-3}.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 3, 3, 3, 6, 6, 6, 6, 6, 6, 2, 2, 5, 5, 5, 5, 5, 2, 2, 6, 6, 6, 6, 6, 6, 2, 2, 2, 2, 4, 4, 4, 4, 3, 3, 3, 5, 5, 5, 5, 5, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 2, 2, 6

Offset: 2

Vladimir Shevelev, May 22 2014

nonn

For the definition of the index of a twin primes pair, see the comment in A242767.
The sequence is constructed as follows. Consider the sequence {A242758(n)-3}. It begins 3,5,11,11,17,17,29,29,29,...
These numbers occur in A001359 (lesser of twin primes) at the indices 1,2,3,3,4,4,5,5,5,...
We add 1: 2,3,4,4,5,5,6,6,6,... (since in A001359 n>=1, while in A242767 n>=2). Now A242767{2,3,4,4,5,5,6,6,6,...} = {1,1,2,2,2,2,3,3,3,...}: we obtain this sequence.

Peter J. C. Moses, Table of n, a(n) for n = 2..5001

Cf. A001359, A006512, A242489, A242490, A242720, A242758, A242767.

More terms from Peter J. C. Moses, May 23 2014

A242720 Smallest even k such that the pair {k-3,k-1} is not a twin prime pair and lpf(k-1) > lpf(k-3) >= prime(n), where lpf = least prime factor (A020639).

Original entry on oeis.org

12, 38, 80, 212, 224, 440, 440, 854, 1250, 1460, 1742, 2282, 2282, 3434, 4190, 4664, 4760, 4760, 6890, 8054, 8054, 8054, 12374, 12830, 12830, 13592, 13592, 14282, 17402, 17402, 18212, 22502, 22502, 22502, 25220, 28202, 28202, 32234, 32402, 32402, 38012

Offset: 2

Vladimir Shevelev, May 21 2014

nonn

The sequence is nondecreasing. See comment in A242758.
a(n) >= prime(n)^2+3. Conjecture: a(n) <= prime(n)^4. - Vladimir Shevelev, Jun 01 2014
Conjecture. There are only a finite number of composite numbers of the form a(n)-1. Peter J. C. Moses found only two: a(16)-1 = 4189 = 59*71 and a(20)-1 = 6889 = 83^2 and no others up to a(2501). Most likely, there are no others. - Vladimir Shevelev, Jun 09 2014

Peter J. C. Moses, Table of n, a(n) for n = 2..2501
V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 [math.GM], 2009-2014, (Section 10).

Cf. A020639, A001359, A006512, A070155, A242489, A242490, A242758, A242847, A246748, A246821, A246824.

lpf[n_] := FactorInteger[n][[1, 1]];
Clear[a]; a[n_] := a[n] = For[k = If[n <= 2, 2, a[n-1]], True, k = k+2, If[Not[PrimeQ[k-3] && PrimeQ[k-1]] && lpf[k-1] > lpf[k-3] >= Prime[n], Return[k]]];
Table[a[n], {n, 2, 50}] (* Jean-François Alcover, Nov 02 2018 *)

lpf(k) = factorint(k)[1,1];
vector(60, n, k=6; while((isprime(k-3) && isprime(k-1)) || lpf(k-1)<=lpf(k-3) || lpf(k-3)Colin Barker, Jun 01 2014

Conjecturally, a(n) ~ (prime(n))^2, as n goes to infinity (cf. A246748, A246821). - Vladimir Shevelev, Sep 02 2014

For n>=3, a(n) >= (prime(n)+1)^2 + 2. Equality holds for terms of A246824. - Vladimir Shevelev, Sep 04 2014

A242847 Numbers n for which A242719(n) > A242720(n).

Original entry on oeis.org

19, 35, 38, 41, 45, 50, 53, 56, 57, 58, 59, 63, 76, 77, 78, 79, 80, 81, 83, 84, 85, 92, 93, 95, 96, 108, 109, 112, 113, 116, 117, 124, 125, 126, 142, 143, 146, 154, 157, 173, 184, 185, 186, 193, 194, 195, 196, 197, 203, 215, 217, 224, 227, 232, 233, 237, 241

Offset: 1

Vladimir Shevelev, Jun 02 2014

nonn

The sequence is infinite, in view of a strong closeness between counting functions of numbers N_1 for which lpf(N_1-3) > lpf(N_1-1) >= prime(n) and numbers N_2 for which lpf(N_2-1) > lpf(N_2-3) >= prime(n), if {N_2-3, N_2-1} is not a pair of twin primes, where p_n=prime(n) and lpf=least prime factor (A020639). (Cf., for example, A243803-A243804). This closeness is explained by a somewhat symmetry (for details, see Shevelev's link).

However, it is very interesting to find an analytical proof of infinity of this and complementory sequences.

V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 [math.GM], 2009-2014 (Section 14).

Cf. A242719, A242720, A242758.

lpf[k_] := FactorInteger[k][[1, 1]];
a19[n_ /; n>1] := a19[n] = For[k = If[n == 2, 10, a19[n-1]], True, k = k+2, If[lpf[k-3] > lpf[k-1] >= Prime[n], Return[k]]];
a20[n_ /; n>1] := a20[n] = For[k = If[n <= 2, 2, a20[n-1]], True, k = k+2, If[Not[PrimeQ[k-3] && PrimeQ[k-1]] && lpf[k-1] > lpf[k-3] >= Prime[n], Return[k]]];
Select[Range[250], a19[#] > a20[#]&] (* Jean-François Alcover, Nov 06 2018 *)

More terms from Peter J. C. Moses, Jun 02 2014

A242807 Smallest lesser of twin primes (A001359) with index n, or a(n)=0, if there are no such twin primes.

Original entry on oeis.org

3, 11, 29, 59, 137, 101, 2309, 1151, 521, 1427, 1229, 419, 5849, 3119, 5417, 10271, 1607, 9629, 1019, 809, 13217, 9239, 15581, 8819, 29021, 84059, 13679, 18911, 14867, 45119, 54401, 60647, 60089, 142589, 78137, 61979, 179381, 26681, 123377, 293861, 89519

Offset: 1

Vladimir Shevelev, May 23 2014

nonn

For definition of index of twin primes pair, see A242767.

We conjecture that, for every n, a(n)>0.

Cf. A001359, A242758, A242767.

More terms from Peter J. C. Moses, May 26 2014

A242881 Positions of lesser of twin primes in A001359 with index 2.

Original entry on oeis.org

3, 4, 10, 12, 14, 15, 23, 32, 33, 37, 39, 50, 58, 60, 64, 66, 82, 86, 90, 91, 93, 111, 112, 114, 128, 139, 143, 155, 157, 158, 162, 167, 171, 179, 190, 197, 198, 199, 207, 223, 226, 231, 241, 248, 255, 262, 270, 280, 282, 286, 293, 306, 313, 317, 318, 325, 327

Offset: 1

Vladimir Shevelev, May 25 2014

nonn

For the definition of the index of a twin primes pair, see the comment in A242767.

Peter J. C. Moses, Table of n, a(n) for n = 1..5000

Cf. A001359, A242758, A242767, A242768, A242807.

More terms from Peter J. C. Moses, May 25 2014

A242913 Positions of smaller of twin primes in A001359 with index 3.

Original entry on oeis.org

5, 6, 8, 17, 19, 29, 38, 44, 45, 46, 52, 54, 55, 65, 67, 71, 78, 80, 87, 95, 96, 97, 103, 106, 113, 119, 121, 124, 135, 136, 138, 152, 166, 187, 188, 191, 192, 208, 209, 212, 217, 237, 253, 254, 259, 269, 271, 275, 277, 288, 300, 308, 316, 320, 331, 349, 355

Offset: 1

Vladimir Shevelev, May 26 2014

nonn

For the definition of the index of a twin prime pair, see the comment in A242767.

Peter J. C. Moses, Table of n, a(n) for n = 1..5000

Cf. A001359, A242758, A242767, A242768, A242807, A242881.

A242767=Join[{1},Differences[Flatten[Position[Differences[Prime[Range[10^4]]],2]]]];
A242913=Flatten[Position[A242767,3]] (* Peter J. C. Moses, May 27 2014 *)

More terms from Peter J. C. Moses, May 26 2014

A243803 Number of numbers k in interval [(p_n)^2+1, (p_n)^4] for which lpf(k-3)>lpf(k-1)>=p_n, where p_n=prime(n) and lpf = A020639.

Original entry on oeis.org

40, 85, 393, 625, 1557, 2106, 4069, 9558, 11476, 22060, 31530, 35998, 49142, 76678, 113799, 125010, 176824, 216378, 234064, 313511, 372054, 481764, 668344, 768307, 811635, 926452, 975785, 1105924, 1751993, 1949976, 2299392, 2394921, 3130534, 3250605, 3751262, 4306910, 4683674, 5332960

Offset: 3

Vladimir Shevelev and Peter J. C. Moses, Jun 10 2014

nonn

a(n) and A243804(n) approximate each other with a small relative error.

Positions n for which a(n) < A243804(n) are 11, 13, 14, 17, 18, 19, 20 (...).

Cf. A020639 (lpf), A242719, A242720, A242847, A242758, A243804.

A243804 Number of numbers k in interval [(p_n)^2+1, (p_n)^4] for which lpf(k-1)>lpf(k-3)>=p_n, such that {k-3, k-1} is not a pair of twin primes, where p_n=prime(n) and lpf = A020639.

Original entry on oeis.org

36, 84, 382, 593, 1526, 2070, 4023, 9536, 11535, 22050, 31552, 36034, 49032, 76464, 113887, 125138, 176940, 216419, 233932, 313011, 371787, 480984, 666608, 767403, 811022, 925567, 974900, 1104796, 1749737, 1948447, 2298322, 2393928, 3129862, 3248932, 3750166, 4305141, 4682343, 5332158

Offset: 3

Vladimir Shevelev and Peter J. C. Moses, Jun 10 2014

nonn

a(n) and A243803(n) approximate each other with the relative error tending to zero with growth of n.

Cf. A020639, A242719, A242720, A242847, A242758, A243803.

A242917 Positions of smaller of twin primes in A001359 with index 4.

Original entry on oeis.org

7, 16, 20, 28, 30, 34, 35, 40, 49, 69, 74, 84, 89, 101, 105, 108, 133, 134, 142, 148, 154, 159, 169, 176, 182, 185, 194, 202, 213, 215, 220, 221, 225, 232, 235, 238, 251, 256, 261, 303, 310, 311, 314, 322, 323, 329, 330, 342, 343, 353, 354, 360, 382, 396, 398

Offset: 1

Vladimir Shevelev, May 26 2014

nonn

For the definition of the index of a twin prime pair, see the comment in A242767.

Peter J. C. Moses, Table of n, a(n) for n = 1..5000

Cf. A001359, A242758, A242767, A242768, A242807, A242881, A242913.

A242767=Join[{1},Differences[Flatten[Position[Differences[Prime[Range[10^4]]],2]]]];
A242917=Flatten[Position[A242767,4]] (* Peter J. C. Moses, May 27 2014 *)

More terms from Peter J. C. Moses, May 26 2014

cp's OEIS Frontend

A242767 Numbers of repetitions of terms in A242758.

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A242768 Indices of twin primes in {A242758(n)-3}.

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A242720 Smallest even k such that the pair {k-3,k-1} is not a twin prime pair and lpf(k-1) > lpf(k-3) >= prime(n), where lpf = least prime factor (A020639).

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A242847 Numbers n for which A242719(n) > A242720(n).

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A242807 Smallest lesser of twin primes (A001359) with index n, or a(n)=0, if there are no such twin primes.

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A242881 Positions of lesser of twin primes in A001359 with index 2.

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A242913 Positions of smaller of twin primes in A001359 with index 3.

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A243803 Number of numbers k in interval [(p_n)^2+1, (p_n)^4] for which lpf(k-3)>lpf(k-1)>=p_n, where p_n=prime(n) and lpf = A020639.

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A243804 Number of numbers k in interval [(p_n)^2+1, (p_n)^4] for which lpf(k-1)>lpf(k-3)>=p_n, such that {k-3, k-1} is not a pair of twin primes, where p_n=prime(n) and lpf = A020639.

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A242917 Positions of smaller of twin primes in A001359 with index 4.

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