A242847 -id:A242847 - cp's OEIS frontend

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

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

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.

A247280 Numbers n for which A242720(n) = prime(n)*(prime(n)+4)+3.

Original entry on oeis.org

4, 6, 8, 19, 50, 59, 63, 65, 78, 85, 93, 112, 117, 143, 237, 254, 264, 276, 287, 303, 333, 371, 380, 425, 435, 440, 447, 459, 483, 485, 537, 612, 614, 659, 731, 851, 877, 920, 983, 994, 1025, 1080, 1096, 1182, 1358, 1380, 1468, 1476, 1481, 1582, 1628, 1690

Offset: 1

Vladimir Shevelev, Sep 11 2014

nonn

prime(n)*(prime(n)+4) + 3, such that prime(n)+4 and prime(n)*(prime(n)+4)+2 are primes, is the second minimal possible value of A242720(n) after (prime(n)+1)^2 + 2, n>=3 (cf. A246824).

Cf. A242719, A242720, A242847, A245363, A246748, A246824, A247011.

More terms from Peter J. C. Moses, Sep 11 2014

A247279 Numbers n such that A242720(n) = prime(n)(prime(n)+4)+3 and A242719(n) - A242720(n) = 2(prime(n)-1).

Original entry on oeis.org

19, 920, 2869, 4704, 8125, 10194, 10939, 17588, 22661, 29856, 31178, 31779, 53624, 59035, 61931, 66944, 72104, 81247, 91456, 98840, 103631, 106187, 117959, 123535, 131824, 133446, 168209, 184888, 189389, 214743, 215352, 218421, 218799, 227088, 237917, 245854

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Sep 11 2014

nonn

The sequence is infinite if there are infinitely many primes p_n such that p_n+4, p_n+6, p_n*(p_n+4)+2, p_n*(p_n+6)-2 are primes, but p_n^2-2 is not prime.

If the sequence A246748 is also infinite, then these two sequences show that the difference A242720(n) - A242719(n) changes its sign infinitely many times.

			If n=920, prime(920)=7207, we have A242720(920) = 7207*7211+3 = 51969680 and A242919(920) -  A242920(920) = 51984092 - 51969680 = 14412 = 2*(prime(920)-1).

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

Cf. A242719, A242720, A242847, A246748.

Intersection of A245363 and A247280.

A247549 Numbers n for which A242720(n) = prime(n)*(prime(n)+8)+3.

Original entry on oeis.org

5, 32, 43, 79, 126, 142, 523, 576, 722, 771, 1026, 1152, 1234, 1402, 1442, 1480, 1623, 1630, 1767, 1814, 1829, 1962, 1995, 2062, 2084, 2353, 2705, 3104, 3174, 3355, 3588, 3718, 4005, 4035, 4126, 4266, 4581, 4616, 4785, 4854, 4859, 5068, 5131, 5145, 5164

Offset: 1

Vladimir Shevelev, Sep 19 2014

nonn

prime(n)*(prime(n)+8) + 3, such that prime(n)+8 and prime(n)*(prime(n)+8)+2 are primes, is the third minimal possible value of A242720(n) after (prime(n)+1)^2 + 2, n>=3 (cf. A246824) and prime(n)*(prime(n)+4) + 3 (cf. A247280).

Cf. A242719, A242720, A242847, A245363, A246748, A246824, A247011, A147280.

If n is in the sequence, then prime(n) == 1 (mod 10), A242720(n) == 12 (mod 100).

More terms from Peter J. C. Moses, Sep 19 2014

cp's OEIS Frontend

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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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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A247280 Numbers n for which A242720(n) = prime(n)*(prime(n)+4)+3.

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A247279 Numbers n such that A242720(n) = prime(n)*(prime(n)+4)+3 and A242719(n) - A242720(n) = 2*(prime(n)-1).

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A247549 Numbers n for which A242720(n) = prime(n)*(prime(n)+8)+3.

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A247279 Numbers n such that A242720(n) = prime(n)(prime(n)+4)+3 and A242719(n) - A242720(n) = 2(prime(n)-1).