A220273 -id:A220273 - cp's OEIS frontend

A218831 a(n) is the least r > 1 for which the interval (rn, r(n+1)) contains no prime, or a(n)=0 if no such r exists.

0, 0, 0, 2, 0, 4, 2, 3, 0, 2, 3, 2, 2, 0, 6, 2, 2, 3, 2, 6, 3, 2, 4, 2, 2, 7, 2, 2, 4, 3, 2, 2, 4, 2, 4, 4, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 4, 3, 2, 3, 4, 2, 3, 2, 2, 2, 2, 2, 2, 4, 2, 5, 2, 2, 3, 3, 2, 2, 2, 2, 4, 4, 2, 2, 3, 2, 2, 3, 2, 4, 2, 2, 3, 2

Offset: 1

Vladimir Shevelev, Charles R Greathouse IV and Peter J. C. Moses, Nov 07 2012

nonn

In the first 50000000 terms a(n) is 0 only for n=1, 2, 3, 5, 9, 14. In the same range the largest value of a(n) is 16 at n=2540, 77384, 1679690, 3240054, 13078899.
a(1)=0 is "Bertrand's postulate," which states that there is always a prime between k and 2*k. This was first proved by P. Chebyshev.
Note that the equations a(2) = a(3) = 0 are results of M. El. Buchraoui and A. Loo respectively and could be proved with the uniform positions, using Theorem 30 for generalized Ramanujan numbers from the Shevelev link. The equation a(5) = 0 follows from the result of J. Nagura. For proof of the equations a(9)=a(14)=0, we used a known result of L. Schoenfeld (1976) that states that for n>2010760, between n and n*(1+1/16597) there is always a prime.

Peter J. C. Moses., Table of n, a(n) for n = 1..20000
N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, Generalized Ramanujan primes, arXiv 2011.
M. El Bachraoui, Primes in the interval [2n,3n], Int. J. Contemp. Math. Sciences 1:13 (2006), pp. 617-621.
N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, Generalized Ramanujan primes, Combinatorial and Additive Number Theory, Springer Proc. in Math. & Stat., CANT 2011 and 2012, Vol. 101 (2014), 1-13
A. Loo, On the primes in the interval [3n,4n], International Journal of Contemporary Mathematical Sciences, volume 6, number 38, pages 1871-1882, 2011.
J. Nagura, On the interval containing at least one prime number, Proc. Japan Acad., 28 (1952), 177-181.
S. Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., 11 (1919), 181-182.
V. Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4
Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, On intervals (kn, (k+1)n) containing a prime for all n>1, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. arXiv:1212.2785
L. Schoenfeld, Sharper bounds for the Chebyshev functions theta(x) and psi(x). II, Math. Comp. 30 (1975) 337-360.

Cf. A218769, A220268, A220269, A220273, A220274, A220281, A220315.

rmax = 100; a[n_] := Catch[ For[r = 2, r <= rmax, r++, If[PrimePi[r*n] == PrimePi[r*(n + 1)], Throw[r], If[r == rmax, Throw[0]]]]]; Table[ a[n] , {n, 1, 87}] (* Jean-François Alcover, Dec 13 2012 *)

a(n) = 0 <=> A220315(k) = n for some k. - Jonathan Sondow, Aug 04 2017

A220274 a(n) is the smallest number such that for all N >= a(n) there are at least n primes between 9N and 10N.

Original entry on oeis.org

2, 14, 23, 23, 34, 36, 57, 58, 60, 60, 77, 86, 100, 100, 102, 123, 149, 149, 149, 149, 187, 187, 200, 200, 200, 202, 209, 227, 234, 268, 269, 269, 270, 319, 319, 331, 332, 332, 333, 345, 347, 350, 350, 353, 359, 360, 377, 401, 421, 440, 449, 479, 479, 487, 491

Offset: 1

Vladimir Shevelev, Charles R Greathouse IV, and Peter J. C. Moses, Dec 09 2012

nonn

Peter J. C. Moses, Table of n, a(n) for n = 1..3000
N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, and J. Sondow, Generalized Ramanujan primes, arXiv:1108.0475 [math.NT], 2011.
N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, and J. Sondow, Generalized Ramanujan primes, Combinatorial and Additive Number Theory, Springer Proc. in Math. & Stat., CANT 2011 and 2012, Vol. 101 (2014), 1-13
Vladimir Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4
Vladimir Shevelev, Charles R. Greathouse IV, and Peter J. C. Moses, On intervals (kn, (k+1)n) containing a prime for all n>1, arXiv:1212.2785 [math.NT], 2012.

Cf. A084140, A220268, A220269, A220273.

a(n) <= ceiling(R_(10/9)(n)/10), where R_v(n) (v>1) are generalized Ramanujan numbers (see Shevelev's link). In particular, for n >= 1, {R_(10/9) (n)} = {127, 223, 227, 269, 349, 359, 569, 587, 593, 739, 809, 857, 991, 1009, ...}. Moreover, if R_(10/9)(n) == 1 (mod 10), then a(n) = ceiling(R_(10/9)(n)/10).

A220281 a(n) is the smallest number, such that for all N >= a(n) there are at least n primes between 14N and 15N.

Original entry on oeis.org

2, 11, 24, 37, 38, 39, 50, 96, 96, 96, 96, 97, 97, 125, 125, 132, 178, 178, 178, 179, 179, 180, 213, 221, 222, 222, 224, 235, 235, 242, 282, 283, 307, 309, 310, 360, 360, 361, 362, 366, 367, 367, 377, 377, 377, 421, 422, 458, 458, 502, 503, 504

Offset: 1

Vladimir Shevelev, Charles R Greathouse IV and Peter J. C. Moses, Dec 09 2012

nonn

Peter J. C. Moses, Table of n, a(n) for n = 1..3000
N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, and J. Sondow, Generalized Ramanujan primes, arXiv:1108.0475 [math.NT], 2011.
N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, and J. Sondow, Generalized Ramanujan primes, Combinatorial and Additive Number Theory, Springer Proc. in Math. & Stat., CANT 2011 and 2012, Vol. 101 (2014), 1-13.
Vladimir Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4.
Vladimir Shevelev, Charles R. Greathouse IV, and Peter J. C. Moses, On intervals (kn, (k+1)n) containing a prime for all n>1, arXiv:1212.2785 [math.NT], 2012.

Cf. A084140, A220268, A220269, A220273, A220274.

a(n) <= ceiling(R_(15/14)(n)/15), where R_v(n) (v>1) are generalized Ramanujan numbers (see Shevelev's link). In particular, for n >= 1, {R_(15/14)(n)}={127, 307, 347, 563, 569, 733, 1423, 1427, 1429, 1433, 1439, 1447, ...}. Moreover, if R_(15/14)(n) == 1 or 2 (mod 10), then a(n) = ceiling(R_(15/14)(n)/15).

cp's OEIS Frontend

A218831 a(n) is the least r > 1 for which the interval (rn, r(n+1)) contains no prime, or a(n)=0 if no such r exists.

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A220274 a(n) is the smallest number such that for all N >= a(n) there are at least n primes between 9N and 10N.

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A220281 a(n) is the smallest number, such that for all N >= a(n) there are at least n primes between 14N and 15N.

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Author

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cp's OEIS Frontend

A218831 a(n) is the least r > 1 for which the interval (r*n, r*(n+1)) contains no prime, or a(n)=0 if no such r exists.

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A220274 a(n) is the smallest number such that for all N >= a(n) there are at least n primes between 9*N and 10*N.

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Author

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A220281 a(n) is the smallest number, such that for all N >= a(n) there are at least n primes between 14*N and 15*N.

Views

Author

Keywords

Links

Crossrefs

Formula

A218831 a(n) is the least r > 1 for which the interval (rn, r(n+1)) contains no prime, or a(n)=0 if no such r exists.

A220274 a(n) is the smallest number such that for all N >= a(n) there are at least n primes between 9N and 10N.

A220281 a(n) is the smallest number, such that for all N >= a(n) there are at least n primes between 14N and 15N.