A242719 -id:A242719 - cp's OEIS frontend

A246748 Numbers n such that A242719(n) = (prime(n))^2+1 and A242720(n) - A242719(n) = 2*(prime(n)+1).

3, 52, 104, 209, 343, 373, 398, 473, 628, 2633, 3273, 7538, 8060, 8813, 9025, 10847, 12493, 13768, 14196, 15486, 16865, 17486, 18362, 18613, 18842, 21175, 23522, 31825, 33537, 34507, 38740, 39603, 41802, 41947, 43314, 45479, 47550, 47668, 47787, 50321, 50682

Offset: 1

Vladimir Shevelev, Sep 02 2014

nonn

If the sequence is infinite, then lim inf(A242719(k)/(prime(k))^2) = 1 and lim inf(A242720(k)/(prime(k))^2) = 1.
In connection with this, one can conjecture that A242719(k) ~ A242720(k) ~ (prime(k))^2, as k goes to infinity (cf. A246819, A246821).
n is in the sequence if and only if prime(n)>=5 and is in the intersection of A001359, A062326, A157468.
Proof. Firstly note that A242719(n) = prime(n)^2 + 1 if and only if prime(n)^2 - 2 is prime. Indeed, let prime(n)^2 + 1 be A242719(n). Then we have lpf(prime(n)^2 - 2) > lpf(prime(n)^2) = prime(n). It is possible only when prime(n)^2 - 2 is prime, i. e., prime(n) is in A062326. Add that prime(n)^2+1 is the smallest value of A242719(n).
Let A242720(n) = A242719(n) + 2*prime(n) + 2 = prime(n)^2 + 2*prime(n) + 3. Then, by the definition of A242720, we have lpf(prime(n)^2 + 2*prime(n) + 2) > lpf(prime(n)*(prime(n)+2)) >= prime(n). Thus prime(n) + 2 is prime, i.e., prime(n) is in A001359. Besides, lpf(prime(n)^2 + 2*prime(n) + 2) > prime(n), or lpf((prime(n)+1)^2 + 1) >= prime(n+1) = prime(n) + 2. So (prime(n)+1)^2+1 is prime, i.e., prime(n) is also in A157468.
Add that, for n>=3, N=prime(n)^2 + 2*prime(n) + 3 is the smallest possible value of A242720(n). Indeed, let prime(n)^2+1 <= N <= prime(n)^2 + 2*prime(n) + 2. Then prime(n)^2-2 <= N - 3 <= prime(n)^2 + 2*prime(n) - 1. Since it should be lpf(N-3) >= prime(n), then there are only two possibilities: N-3 = prime(n)^2 + prime(n) or N-3 = prime(n)^2. However, lpf(prime(n)^2 + prime(n)) = 2, while, although lpf(prime(n)^2) = prime(n), however, in this case, lpf(N-1) = lpf(prime(n)^2+2) = 3, n>=3, and, so the inequalities lpf(N-1) > lpf(N-3) >= prime(n) are impossible in the considered cases for n>=3. - Vladimir Shevelev, Sep 03 2014

Cf. A001359, A062326, A157468, A242719, A242720, A246819, A246821.

More terms from Peter J. C. Moses, Sep 02 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

A246819 Max_{2<=k<=n} floor(A242719(k)/prime(k)) - prime(n).

Original entry on oeis.org

0, 0, 0, 4, 2, 4, 2, 13, 7, 13, 7, 4, 2, 0, 17, 11, 9, 6, 2, 24, 21, 17, 11, 12, 8, 6, 2, 18, 29, 15, 11, 5, 3, 16, 14, 8, 18, 14, 8, 22, 20, 10, 30, 29, 27, 15, 3, 6, 8, 4, 0, 30, 20, 14, 60, 54, 52, 46, 42, 40, 30, 16, 12, 10, 41, 27, 21, 11, 20, 16, 10, 6

Offset: 2

Vladimir Shevelev, Sep 04 2014

nonn

Conjecture: a(n) = o(prime(n)), as n goes to infinity.

If the conjecture is true, then A242719(n) ~ prime(n)^2. Indeed, A242719(n) >= prime(n)^2 + 1; on the other hand, by the conjecture, we have A242719(n)/prime(n) <= a(n) + 1 + prime(n) = prime(n)*(1+o(1)).

Peter J. C. Moses, Table of n, a(n) for n = 2..2001

Cf. A242719, A242720, A246748, A246821.

More terms from Peter J. C. Moses, Sep 04 2014

A247011 Numbers n for which A242719(n) = (prime(n) + 2)^2 + 1.

Original entry on oeis.org

5, 7, 13, 17, 26, 33, 64, 81, 98, 140, 171, 176, 190, 201, 215, 225, 318, 332, 336, 444, 469, 475, 495, 551, 558, 563, 577, 601, 636, 671, 828, 849, 862, 870, 948, 1004, 1064, 1074, 1189, 1198, 1230, 1238, 1305, 1328, 1445, 1449, 1528, 1618, 1634, 1642, 1679

Offset: 1

Vladimir Shevelev, Sep 09 2014

nonn

(prime(n) + 2)^2 + 1 is the second minimal possible value of A242719(n) after prime(n)^2 + 1. Indeed, by the definition lpf(A242719(n) - 3) > lpf(A242719(n) - 1) >= prime(n), thus after prime(n)^2 + 1 we should consider prime(n)*(prime(n) + 2) + 1. Then prime(n) should be lesser number of twin primes, but then prime(n) + 1 == 0 (mod 3). So, prime(n)*(prime(n) + 2) - 2 == 0 (mod 3). Analogously one can prove that prime(n)*(prime(n) + 4) - 2 == 0 (mod 3).

Note that for the sequence prime(n+1) is in intersection of A006512 and A062326, but prime(n) is not in A062326.

François Marques, Table of n, a(n) for n = 1..109

Cf. A242719, A246748.

If prime(n) is not in A062326, then A242719(n) >= (prime(n)+2)^2 + 1.

Intersection of A247011 and A246824 forms sequence 81, 215, 828, 1189, 1634, ... For these values of n we have A242719(n) - A242720(n) = 2*(prime(n) + 1).

More terms from Peter J. C. Moses, Sep 09 2014

A245363 Numbers n for which A242719(n) = prime(n)*(prime(n)+6) + 1.

Original entry on oeis.org

19, 37, 49, 69, 73, 102, 165, 236, 253, 365, 465, 542, 595, 694, 713, 723, 762, 920, 962, 979, 1119, 1162, 1259, 1334, 1387, 1441, 1706, 1797, 1843, 1906

Offset: 1

Vladimir Shevelev, Sep 09 2014

nonn

These are such numbers n for which prime(n), prime(n)+6, prime(n)*(prime(n)+6)-2 are primes, but prime(n) is not in A062326; besides, if prime(n) is in A001359, then also prime(n+1) is not in A062326.

Prime(n)*(prime(n)+6) + 1 is the third minimal possible value of A242719(n) after prime(n)^2 + 1 and (prime(n)+2)^2 + 1 (cf. A247011 and comment there).

Cf. A242719, A246748, A247011.

prime(n) == 7 (mod 10), A242719(n) == 92 (mod 100).

More terms from Peter J. C. Moses, Sep 09 2014

A243960 Primes that do not divide the numbers of the form A242719(n)-1.

Original entry on oeis.org

2, 11, 17, 23, 31, 41, 53, 59, 79, 83, 101, 113, 137, 149, 167, 179, 197, 229, 251, 263, 269, 281, 283, 311, 317, 331, 349, 383, 433, 439, 457, 461, 463, 479, 487, 499, 503, 509, 521, 547, 577, 593, 599, 619, 641, 661, 673, 683, 691, 709, 739, 769, 787, 809

Offset: 1

Vladimir Shevelev, Jun 16 2014

nonn

If a prime p=prime(k) does not divide A242719(i)-1 for i=2,3,...,k, then it is in the sequence.

Cf. A242719.

More terms from Peter J. C. Moses, Jun 17 2014

A246501 a(n) is the smallest term of A242719 that is repeated exactly n times, or 0 if there is no such term.

Original entry on oeis.org

10, 170, 3722, 85040, 273530, 9010484, 3659570, 22972850, 1875409412, 2023830170

Offset: 1

Vladimir Shevelev, Aug 27 2014

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

Integer parts of sqrt(a(n)-1) are 3,13,61,291,523,3001,1913,4793,... (as for all terms of A242719, if a(n)-1 is perfect square, then sqrt(a(n)-1) is prime).

Cf. A246502, A242719, A242720.

More terms from Peter J. C. Moses, Aug 27 2014

a(9)-a(10) from Jinyuan Wang, Mar 13 2020

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.

A243199 Let A242719(n)-1 = prime(l)*prime(m), l<=m; a(n)=m-l.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 2, 1, 4, 0, 0, 4, 4, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 4, 4, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 4, 5, 4, 4, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 2

Vladimir Shevelev, Jun 01 2014

nonn

Cf. A242719, A243154, A243158.

More terms from Peter J. C. Moses, Jun 01 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

cp's OEIS Frontend

A246748 Numbers n such that A242719(n) = (prime(n))^2+1 and A242720(n) - A242719(n) = 2*(prime(n)+1).

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

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A246819 Max_{2<=k<=n} floor(A242719(k)/prime(k)) - prime(n).

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A247011 Numbers n for which A242719(n) = (prime(n) + 2)^2 + 1.

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A245363 Numbers n for which A242719(n) = prime(n)*(prime(n)+6) + 1.

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A243960 Primes that do not divide the numbers of the form A242719(n)-1.

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A246501 a(n) is the smallest term of A242719 that is repeated exactly n times, or 0 if there is no such term.

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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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A243199 Let A242719(n)-1 = prime(l)*prime(m), l<=m; a(n)=m-l.

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