A242720 -id:A242720 - cp's OEIS frontend

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

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

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.

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

A242974 Let M_n = A002110(n) (the n-th primorial), let N*(n)(N**(n), respectively) be the number of numbers k in [1, M_n] for which lpf(k-3) > lpf(k-1) >= prime(n) (lpf(k-1) > lpf(k-3) >= prime(n), respectively) such that k-3, k-1 are not twin primes, where lpf=least prime factor. Then a(n) = N*(n) - N**(n).

Original entry on oeis.org

1, 1, 3, 25, 67, 131, 1556, -1671

Offset: 3

Vladimir Shevelev, Jun 13 2014

Small values of |a(n)| with respect to N*(n) + N**(n) (cf. A243867) clearly demonstrate the fact of statistical closeness of N*(n) and N**(n). See also comment in A243867.

If we don't exclude twin primes in the definition then, instead of this sequence, we would obtain -3, -14, -66, -443, -4569, -57422, -894506, -18465384, ... (cf. A000882). Thus twin primes strongly destroy the statistical closeness of N*(n) and N**(n).

Vladimir Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 [math.GM], 2009-2014 (Sections 10,14).

Cf. A020639, A243867, A242719, A242720, A242758, A243803, A243804.

lpf(k) = factorint(k)[1, 1];
a(n) = {my(p=prime(n), r=1, s=2, t, u=0); for(k=4, prod(i=1, n, prime(i)), if((t=lpf(k-1))>r, if(r>=p&&(r=p, u++)); r=s; s=t); u; } \\ Jinyuan Wang, Mar 13 2020

More terms from Peter J. C. Moses, Jun 13 2014

A243867 Sum of the numbers N*(n) and N**(n) in A242974.

Original entry on oeis.org

1, 7, 97, 1289, 20611, 365775, 7813466, 212149365

Offset: 3

Vladimir Shevelev, Jun 13 2014

V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 (Sections 10,14)

Cf. A242974, A242719, A242720, A242758, A243803, A243804.

Let B(n) be the number of twin primes pairs not exceeding the n-th primorial M_n = A002110(n). Then we know that B(n) = O(M_n/(log(M_n))^2) = o(M_n/log((p_(n-1)))^2. For sufficiently large n, a(n) + B(n) >= 0.416...*M_n/(log(prime(n-1)))^2 (cf. Shevelev link) and thus for large n, for example, we have a(n) >= 0.4*M_n/(log(prime(n-1)))^2.

More terms from Peter J. C. Moses, Jun 13 2014

A244412 Least even k such that sfdf(k-1) > sfdf(k-3) >= A050376(n), where sfdf(n) is the smallest Fermi-Dirac factor of n (A223490), and k-3 is not the lesser of a pair of Fermi-Dirac twin primes (A229064).

Original entry on oeis.org

18, 38, 38, 80, 102, 212, 224, 440, 440, 440, 578, 728, 1250, 1460, 1742, 2012, 2282, 3434, 3482, 4190, 4664, 4760, 4760, 6890, 7212, 7212, 7212, 8054, 10772, 12830, 12830, 13592, 13592, 14282, 17402, 17402, 17402, 18212, 22502, 22502, 22502, 25220, 28202

Offset: 2

Vladimir Shevelev and Peter J. C. Moses, Jun 27 2014

nonn

A Fermi-Dirac analog of A242720.

			If k>=4 is even such that k-3 is either 1 or in A050376, then k cannot be a solution. Thus, if n=2, then k=4,6,8,10,12,14 are not allowed; for k=16 we have sfdf(16-1) = 3 < sfdf(16-3) = 13; finally, for k=18 we have sfdf(18-1) = 17 > sfdf(18-3) = 3 = A050376(2). Since 15 is not in A229064,  then a(2)=18.

Cf. A050376, A223490, A242719, A244343, A242720, A229064.

cp's OEIS Frontend

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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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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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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A243867 Sum of the numbers N*(n) and N**(n) in A242974.

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A244412 Least even k such that sfdf(k-1) > sfdf(k-3) >= A050376(n), where sfdf(n) is the smallest Fermi-Dirac factor of n (A223490), and k-3 is not the lesser of a pair of Fermi-Dirac twin primes (A229064).

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