A246821 -id:A246821 - cp's OEIS frontend

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).

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

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

Original entry on oeis.org

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

A246824 Numbers k for which A242720(k) = (prime(k)+1)^2 + 2.

Original entry on oeis.org

3, 35, 41, 52, 57, 81, 104, 209, 215, 343, 373, 398, 473, 477, 584, 628, 768, 774, 828, 872, 1117, 1145, 1189, 1287, 1324, 1435, 1615, 1634, 1653, 1704, 1886, 1925, 2070, 2075, 2123, 2171, 2193, 2425, 2449, 2605, 2633, 2934, 2948, 3019, 3194, 3273, 3533, 3552, 3685, 3758

Offset: 1

Vladimir Shevelev, Sep 04 2014

nonn

By a comment in A246748, A242720(k) >= (prime(k)+1)^2 + 2, and equality is attained in this sequence.

Prime(a(n)) >= 5 and is in the intersection of A001359 and A157468.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A001359, A157468, A242719, A242720, A246748, A246819, A246821.

lpf[n_] := FactorInteger[n][[1, 1]]; aQ[n_] := Module[{k=6}, While[PrimeQ[k-3] && PrimeQ[k-1] || lpf[k-1]<=lpf[k-3] || lpf[k-3]Amiram Eldar, Dec 10 2018 *)

lpf(k) = factorint(k)[1, 1];
f(n) = my(k=6); while((isprime(k-3) && isprime(k-1)) || lpf(k-1)<=lpf(k-3) || lpf(k-3)A242720
isok(n) = f(n) == (prime(n)+1)^2 + 2; \\ Michel Marcus, Dec 10 2018

from sympy import prime, isprime, factorint
A246824_list = [a for a, b in ((n, prime(n)+1) for n in range(3,10**3)) if (not (isprime(b**2-1) and isprime(b**2+1)) and (min(factorint(b**2+1)) > min(factorint(b**2-1)) >= b-1))] # Chai Wah Wu, Jun 03 2019

a(40)-a(50) from b-file by Robert Price, Sep 08 2019

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

cp's OEIS Frontend

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

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A246824 Numbers k for which A242720(k) = (prime(k)+1)^2 + 2.

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

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