A243804 -id:A243804 - cp's OEIS frontend

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

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

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.

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

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

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