A242490 -id:A242490 - cp's OEIS frontend

A242770 Composite numbers of the form A242490(n) - 1.

4189, 12193, 641941, 8778073, 26822599, 11488195489

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 22 2014

a(7) (if it exists) = A242490(k) - 1, k >= 5 * 10^7. - Hiroaki Yamanouchi, Sep 22 2014

			a(1) = A242490(16) - 1 = 4189 = 59 * 71;
a(2) = A242490(21) - 1 = 12193 = 89 * 137;
a(3) = A242490(125) - 1 = 641941 = 727 * 883;
a(4) = A242490(417) - 1 = 8778073 = 2927 * 2999;
a(5) = A242490(675) - 1 = 26822599 = 5081 * 5279;
a(6) = A242490(10171) - 1 = 11488195489 = 107183 * 107183.

Cf. A242489, A242490, A242716.

a(6) from Hiroaki Yamanouchi, Sep 22 2014

A242719 Smallest even k such that lpf(k-3) > lpf(k-1) >= prime(n), where lpf=least prime factor (A020639).

Original entry on oeis.org

10, 26, 50, 170, 170, 362, 362, 842, 842, 1370, 1370, 1850, 1850, 2210, 3722, 3722, 3722, 4892, 5042, 7082, 7922, 7922, 7922, 10610, 10610, 10610, 11450, 13844, 16130, 16130, 17162, 19322, 19322, 24614, 24614, 25592, 29504, 29930, 29930, 36020, 36020

Offset: 2

Vladimir Shevelev, May 21 2014

nonn

The sequence is connected with a sufficient condition for the existence of an infinity of twin primes. In contrast to A242489, this sequence is nondecreasing.
All even numbers of the form A062326(n)^2 + 1 are in the sequence. All a(n)-1 are semiprimes. - Vladimir Shevelev, May 24 2014
a(n) <= A242489(n); a(n) >= prime(n)^2+1. Conjecture: a(n) <= prime(n)^4. - Vladimir Shevelev, Jun 01 2014
Conjecture: all numbers a(n)-3 are primes. Peter J. C. Moses verified this conjecture up to a(2001) (cf. with conjecture in A242720). - Vladimir Shevelev, Jun 09 2014

Jinyuan Wang, Table of n, a(n) for n = 2..5000 (terms 2..2001 from Peter J. C. Moses).

Cf. A001359, A006512, A062326, A137291, A242489, A242490, A242847, A243960, A245363, A246501, A246748, A246819, A247011.

lpf[k_] := FactorInteger[k][[1, 1]];
a[n_] := a[n] = For[k = If[n == 2, 10, a[n-1]], True, k = k+2, If[lpf[k-3] > lpf[k-1] >= Prime[n], Return[k]]];
Array[a, 50, 2] (* Jean-François Alcover, Nov 06 2018 *)

lpf(k) = factorint(k)[1,1];
vector(50, n, k=6; while(lpf(k-3)<=lpf(k-1) || lpf(k-1)Colin Barker, Jun 01 2014

Conjecturally, a(n) ~ (prime(n))^2, as n goes to infinity (cf. A246748, A246819). - Vladimir Shevelev, Sep 02 2014

a(n) = prime(n)^2 + 1 for and only for numbers n>=2 which are in A137291. - Vladimir Shevelev, Sep 04 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

A242758 Smallest even k such that lpf(k-1) > lpf(k-3) >= prime(n), where lpf=least prime factor (A020639).

Original entry on oeis.org

6, 8, 14, 14, 20, 20, 32, 32, 32, 44, 44, 44, 62, 62, 62, 62, 74, 74, 74, 104, 104, 104, 104, 104, 104, 110, 110, 140, 140, 140, 140, 140, 152, 152, 182, 182, 182, 182, 182, 182, 194, 194, 200, 200, 230, 230, 230, 230, 242, 242, 242, 272, 272, 272, 272, 272

Offset: 2

Vladimir Shevelev, May 22 2014

nonn

This is a version of A242720 with the absolute minima of k in the definition. The sequence is nondecreasing. Hypothetically, every pair {a(n)-3, a(n)-1} is a pair of twin primes.
If there exist infinitely many n such that a(n) < A242719(n) < a(n)^2, then from the result in the Shevelev link, it follows that for such n the set of numbers {even k: lpf(k-1) > lpf(k-3) >= prime(n)} either attains the absolute minimum of a(n) only in the case when {a(n)-3, a(n)-1} are twin primes, or does not attain it at all. Therefore, if there is only a finite number of twin primes, we have a contradiction. Thus the above condition is sufficient for infinity of twin primes.
Note also that, if there is only a finite number of twin primes, then after the last pair of them, this sequence will coincide with A242720. Then, in order to avoid a contradiction (again according to the Shevelev link), we should accept that there exists a number N_0 such that, for every n >= N_0, the following inequality holds: max(A242719(n),A242720(n)) > (min(A242719(n),A242720(n)))^2. - Vladimir Shevelev, May 24 2014
It is easy to prove that min(A242719(n), A242720(n)) >= prime(n)^2+1, while we conjecture that max(A242719(n), A242720(n)) <= prime(n)^4. Thus this conjecture implies there are infinitely many twin primes. - Vladimir Shevelev, Jun 01 2014

Peter J. C. Moses, Table of n, a(n) for n = 2..10001
V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 [math.GM], 2009-2014.

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

lpf[k_] := FactorInteger[k][[1, 1]];
a[n_] := a[n] = For[k = If[n == 2, 2, a[n-1]], True, k = k+2, If[lpf[k-1] > lpf[k-3] >= Prime[n], Return[k]]];
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Nov 03 2018 *)

lpf(k) = factorint(k)[1,1];
vector(100, n, k=6; while(lpf(k-1)<=lpf(k-3) || lpf(k-3)Colin Barker, Jun 01 2014

A242767 Numbers of repetitions of terms in A242758.

Original entry on oeis.org

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

A242716 Composite numbers of the form A242489(n) - 3.

Original entry on oeis.org

12317, 50637287, 147111617, 5174048597

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 21 2014

a(5) (if it exists) = A242489(k) - 3, k >= 5 * 10^7. - Hiroaki Yamanouchi, Sep 22 2014

			a(1) = A242489(25) - 3 = 12317 = 109 * 113;
a(2) = A242489(900) - 3 = 50637287 = 7103 * 7129;
a(3) = A242489(1436) - 3 = 147111617 = 12097 * 12161;
a(4) = A242489(7084) - 3 = 5174048597 = 71789 * 72073.

Cf. A242489, A242490, A242770.

a(4) from Hiroaki Yamanouchi, Sep 22 2014

cp's OEIS Frontend

A242770 Composite numbers of the form A242490(n) - 1.

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A242719 Smallest even k such that lpf(k-3) > lpf(k-1) >= prime(n), where lpf=least prime factor (A020639).

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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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A242758 Smallest even k such that lpf(k-1) > lpf(k-3) >= prime(n), where lpf=least prime factor (A020639).

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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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A242716 Composite numbers of the form A242489(n) - 3.

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