A242758 -id:A242758 - cp's OEIS frontend

A242918 Positions of smaller of twin primes in A001359 with index 5.

11, 18, 21, 27, 43, 48, 62, 77, 107, 109, 110, 118, 131, 145, 172, 201, 216, 258, 260, 265, 289, 294, 301, 307, 315, 319, 340, 350, 363, 365, 367, 381, 442, 449, 451, 453, 491, 496, 500, 515, 522, 528, 540, 559, 569, 581, 603, 613, 616, 620, 623, 630, 659, 689

Offset: 1

Vladimir Shevelev, May 26 2014

nonn

For the definition of the index of a twin prime pair, see the comment in A242767.

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

Cf. A001359, A242758, A242767, A242768, A242807, A242881, A242913, A242917.

A242767=Join[{1},Differences[Flatten[Position[Differences[Prime[Range[10^4]]],2]]]];
A242918=Flatten[Position[A242767,5]] (* Peter J. C. Moses, May 27 2014 *)

More terms from Peter J. C. Moses, May 26 2014

A242919 Positions of smaller of twin primes in A001359 with index 6.

Original entry on oeis.org

9, 13, 24, 26, 53, 59, 61, 63, 76, 88, 94, 100, 104, 115, 126, 146, 156, 160, 184, 203, 206, 210, 224, 229, 240, 266, 276, 279, 298, 309, 333, 338, 352, 386, 406, 415, 431, 435, 444, 450, 469, 473, 508, 525, 529, 535, 537, 546, 550, 576, 580, 615, 633, 634

Offset: 1

Vladimir Shevelev, May 26 2014

nonn

For the definition of the index of a twin prime pair, see the comment in A242767.

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

Cf. A001359, A242758, A242767, A242768, A242807, A242881, A242913, A242917, A242918.

A242767=Join[{1},Differences[Flatten[Position[Differences[Prime[Range[10^4]]],2]]]];
A242919=Flatten[Position[A242767,6]] (* Peter J. C. Moses, May 27 2014 *)

More terms from Peter J. C. Moses, May 26 2014

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

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A242918 Positions of smaller of twin primes in A001359 with index 5.

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A242919 Positions of smaller of twin primes in A001359 with index 6.

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

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