A242720 -id:A242720 - 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

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

A246821 Max _{2<=k<=n} floor(A242720(k)/prime(k)) - prime(n).

Original entry on oeis.org

1, 2, 4, 8, 6, 8, 6, 14, 14, 16, 10, 14, 12, 26, 26, 20, 18, 12, 26, 37, 31, 27, 50, 42, 38, 36, 32, 30, 41, 27, 23, 27, 25, 15, 16, 22, 16, 26, 20, 14, 29, 19, 34, 30, 40, 40, 28, 24, 22, 18, 12, 10, 20, 20, 14, 8, 16, 10, 26, 41, 31, 17, 13, 11, 45, 31, 47

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 A242720(n) ~ prime(n)^2. Indeed, A242720(n) >= prime(n)^2 + 2*prime(n) + 3; on the other hand, by the conjecture, we have A242720(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..2501

Cf. A242719, A242720, A246748, A246819.

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

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

A246502 a(n) is the smallest term of A242720 that is repeated exactly n times, or 0 if there is no such term.

Original entry on oeis.org

12, 440, 8054, 129554, 227432, 7986230, 62015624, 280729964

Offset: 1

Vladimir Shevelev, Aug 27 2014

We conjecture that a(n)>0.

Cf. A246501, A242719, A242720.

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

A243200 Let A242720(n)-3 = prime(l)*prime(m), l<=m; a(n)=m-l.

Original entry on oeis.org

0, 1, 1, 3, 1, 1, 1, 3, 4, 4, 3, 2, 2, 6, 6, 5, 1, 1, 5, 2, 2, 2, 10, 5, 5, 3, 3, 3, 2, 2, 2, 1, 1, 1, 3, 2, 2, 5, 1, 1, 3, 3, 3, 3, 6, 7, 4, 1, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 2, 2, 2, 2, 2, 2, 8, 6, 6, 6, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 4, 2, 2

Offset: 2

Vladimir Shevelev, Jun 01 2014

nonn

Since for a prime p>3, p^2 == 1 (mod 3), then for n>2, A242720(n) is not equal to p^2 + 3 (otherwise, lpf(A242720(n) - 1) = 3). So, the sequence has only zero term a(2).

Cf. A242720, A242719, A243154, A243158.

More terms from Peter J. C. Moses, Jun 01 2014

A243990 Primes which do not divide the numbers of the form A242720(n)-3.

Original entry on oeis.org

2, 41, 61, 103, 113, 157, 227, 263, 283, 337, 349, 373, 383, 389, 431, 433, 449, 457, 557, 563, 577, 601, 631, 641, 677, 683, 691, 733, 751, 857, 881, 883, 911, 929, 953, 967, 983, 991, 1009, 1039, 1093, 1097, 1151, 1181, 1279, 1303, 1373, 1427, 1451, 1481

Offset: 1

Vladimir Shevelev, Jun 17 2014

nonn

If a prime p=prime(k) does not divide A242720(i)-3 for i=2,3,...,k, then it is in the sequence.

Cf. A242720, A242719, A243960.

More terms from Peter J. C. Moses, Jun 18 2014

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

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

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A246821 Max _{2<=k<=n} floor(A242720(k)/prime(k)) - prime(n).

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

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A246502 a(n) is the smallest term of A242720 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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A243200 Let A242720(n)-3 = prime(l)*prime(m), l<=m; a(n)=m-l.

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A243990 Primes which do not divide the numbers of the form A242720(n)-3.

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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).