A220293 -id:A220293 - cp's OEIS frontend

A220462 Chebyshev numbers C_v(n) for v=3/2: a(n) is the smallest number such that if x>=a(n), then theta(x)-theta(2x/3)>=nlog(x), where theta(x) = sum_{prime p<=x} log p.

13, 37, 41, 67, 73, 97, 127, 137, 173, 179, 181, 211, 229, 239, 263, 307, 311, 347, 367, 379, 431, 433, 443, 449, 479, 487, 541, 563, 587, 599, 607, 641, 643, 673, 739, 757, 787, 797, 809, 823, 827, 859, 937, 967, 997, 1019, 1031, 1039, 1049, 1061, 1087

Offset: 1

Vladimir Shevelev, Charles R Greathouse IV and Peter J. C. Moses, Dec 15 2012

nonn

All terms are primes.

Up to a(97)=2333, only four terms of the sequence (a(33)=643, a(34)=673, a(76)=1721 and a(77)=1741) are not (3/2)-Ramanujan numbers as in Shevelev's link; up to 2333, the only (3/2)-Ramanujan numbers missing from the sequence are 2, 617, 653, 709, 1709, 1733, and 1747.

N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, Generalized Ramanujan primes, arXiv 2011.
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
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

Cf. A220293.

(* Assuming range of x is from a(n) to 2*a(n) *) theta[x_] := Sum[Log[p], {p, Table[Prime[k], {k, 1, PrimePi[x]}]}]; Clear[a]; a[0] = 2; a[n_] := a[n] = (t = Table[{an, x >= an && theta[x] - theta[2*(x/3)] >= n*Log[x]}, {an, a[n - 1], Prime[4*(n + 1)]}, {x, an, 2*an}]; sp = t // Flatten[#, 1] & // Sort // Split[#, #1[[1]] == #2[[1]] &] &; Select[sp, And @@ (#[[All, 2]]) &] // First // First // First); Table[Print[a[n]]; a[n], {n, 1, 51}] (* Jean-François Alcover, Jan 24 2013 *)

a(n)<=prime(4*(n+1)).

A220463 Chebyshev numbers C_v(n) for v=1.2: a(n) is the smallest number such that if x>=a(n), then theta(x)-theta(5x/6)>=nlog(x), where theta(x)=sum_{prime p<=x}log p.

Original entry on oeis.org

59, 137, 139, 149, 223, 241, 347, 353, 383, 389, 563, 569, 593, 613, 631, 641, 821, 823, 853, 929, 937, 1009, 1013, 1061, 1069, 1277, 1279, 1361, 1427, 1433, 1481, 1487, 1597, 1601, 1607, 1609, 1613, 1973, 1979, 1997, 2011, 2081, 2083, 2113, 2203, 2269, 2273, 2297

Offset: 1

Vladimir Shevelev, Charles R Greathouse IV and Peter J. C. Moses, Dec 15 2012

nonn

All terms are primes.

Up to a(98)=5381, all terms are 1.2-Ramanujan numbers as in Shevelev's link; up to 5381, the only missing 1.2-Ramanujan numbers are 29 and 5171.

N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, Generalized Ramanujan primes, arXiv 2011.
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
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

Cf. A220293, 220462.

k=5; xs=Table[{m,Ceiling[x/.FindRoot[(x (-1300+Log[x]^4))/Log[x]^5==(k+1) m,{x,f[(k+1) m]-1},AccuracyGoal->Infinity,PrecisionGoal->20,WorkingPrecision->100]]},{m,1,101}]; Table[{m,1+NestWhile[#-1&,xs[[m]][[2]],(1/Log[#1]Plus@@Log[Select[Range[Floor[(k #1)/(k+1)]+1,#1],PrimeQ]]&)[#]>m&]},{m,1,100}] (* Peter J. C. Moses, Dec 20 2012 *)

a(n)<=prime(11*(n+1)).

A220474 Chebyshev numbers C_v(n) for v=10/9: a(n) is the smallest number such that if x>=a(n), then theta(x)-theta(9x/10)>=nlog(x), where theta(x)=sum_{prime p<=x}log p.

Original entry on oeis.org

223, 227, 269, 349, 359, 569, 587, 593, 739, 809, 857, 991, 1009, 1019, 1259, 1481, 1483, 1487, 1489, 1861, 1867, 1993, 1997, 2003, 2027, 2267, 2269, 2657, 2671, 2687, 2689, 2699, 3181, 3187, 3307, 3313, 3319, 3323, 3457, 3461, 3491, 3527, 3529, 3581, 3623, 3769, 4049, 4201, 4391, 4481

Offset: 1

Vladimir Shevelev, Charles R Greathouse IV and Peter J. C. Moses, Dec 15 2012

nonn

All terms are primes.

Up to a(99)=9029, all terms are (10/9)-Ramanujan numbers as in Shevelev's link; up to 9029, the only missing (10/9)-Ramanujan number is 127.

N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, Generalized Ramanujan primes, arXiv 2011.
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
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

Cf. A220293, A220462, A220463.

k=9; xs=Table[{m,Ceiling[x/.FindRoot[(x (-1300+Log[x]^4))/Log[x]^5==(k+1) m,{x,f[(k+1) m]-1},AccuracyGoal->Infinity,PrecisionGoal->20,WorkingPrecision->100]]},{m,1,101}]; Table[{m,1+NestWhile[#-1&,xs[[m]][[2]],(1/Log[#1]Plus@@Log[Select[Range[Floor[(k #1)/(k+1)]+1,#1],PrimeQ]]&)[#]>m&]},{m,1,100}] (* Peter J. C. Moses, Dec 20 2012 *)
(* Assuming range of x is from a(n) to 2*a(n) *) Clear[a, theta]; theta[x_] := theta[x] = Sum[Log[p], {p, Table[Prime[k], {k, 1, PrimePi[x]}]}] // N; a[0] = 211(* just to speed-up computation *); a[n_] := a[ n] = (t = Table[an = Prime[pi]; Table[{an, x >= an && theta[x] - theta[9*x/10] >= n*Log[x]}, {x, an, 2*an}], {pi, PrimePi[a[n-1]], 31*(n+1)}]; sp = t // Flatten[#, 1]& // Sort // Split[#, #1[[1]] == #2[[1]]& ]&; Select[sp, And @@ (#[[All, 2]]) &] // First // First // First); Table[Print[a[n]]; a[n], {n, 1, 50}] (* Jean-François Alcover, Feb 11 2013 *)

a(n)<=prime(31*(n+1)).

More terms from Jean-François Alcover, Feb 11 2013

A220475 Chebyshev numbers C_v(n) for v=15/14: a(n) is the smallest number such that if x>=a(n), then theta(x)-theta(14x/15)>=nlog(x), where theta(x)=sum_{prime p<=x} log p.

Original entry on oeis.org

307, 347, 563, 569, 733, 821, 1427, 1429, 1433, 1439, 1447, 1481, 1867, 1931, 1973, 2657, 2659, 2663, 2671, 2683, 3187, 3191, 3313, 3319, 3323, 3461, 3511, 3517, 4001, 4217, 4231, 4597, 4621, 4783, 5387, 5393, 5413, 5417, 5477, 5501, 5639, 5641, 5651, 6067, 6311, 6823, 6857, 7477, 7523, 7537

Offset: 1

Vladimir Shevelev, Charles R Greathouse IV and Peter J. C. Moses, Dec 15 2012

nonn

All terms are primes.

Up to a(100)=15013, all terms are (15/14)-Ramanujan numbers as in Shevelev's link, except for 821; the sequence is missing (15/4)-Ramanujan numbers 127 and 1423 and no others up to 15013.

N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, Generalized Ramanujan primes, arXiv 2011.
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
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

Cf. A220293, A220462, A220463, A220474.

k=14; xs=Table[{m,Ceiling[x/.FindRoot[(x (-1300+Log[x]^4))/Log[x]^5==(k+1) m,{x,f[(k+1) m]-1},AccuracyGoal->Infinity,PrecisionGoal->20,WorkingPrecision->100]]},{m,1,101}]; Table[{m,1+NestWhile[#-1&,xs[[m]][[2]],(1/Log[#1]Plus@@Log[Select[Range[Floor[(k #1)/(k+1)]+1,#1],PrimeQ]]&)[#]>m&]},{m,1,100}] (* Peter J. C. Moses, Dec 20 2012 *)
(* Assuming range of x is from a(n) to 2*a(n) *) Clear[a, theta]; theta[x_] := theta[x] = Sum[Log[p], {p, Table[Prime[k], {k, 1, PrimePi[x]}]}] // N; a[0] = 293(* just to speed-up computation *); a[6] = 821(* the exception noted in comments *); a[n_] := a[ n] = (t = Table[an = Prime[pi]; Table[{an, x >= an && theta[x] - theta[14*x/15] >= n*Log[x]}, {x, an, 2*an}], {pi, PrimePi[a[n - 1]], 32*(n+1)}]; sp = t // Flatten[#, 1] & // Sort // Split[#, #1[[1]] == #2[[1]] &] &; Select[sp, And @@ (#[[All, 2]]) &] // First // First // First); Table[Print[a[n]]; a[n], {n, 1, 50}] (* Jean-François Alcover, Feb 11 2013 *)

a(n) <= prime(32*(n+1)).

More terms from Jean-François Alcover, Feb 11 2013

cp's OEIS Frontend

A220462 Chebyshev numbers C_v(n) for v=3/2: a(n) is the smallest number such that if x>=a(n), then theta(x)-theta(2*x/3)>=n*log(x), where theta(x) = sum_{prime p<=x} log p.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A220463 Chebyshev numbers C_v(n) for v=1.2: a(n) is the smallest number such that if x>=a(n), then theta(x)-theta(5*x/6)>=n*log(x), where theta(x)=sum_{prime p<=x}log p.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A220474 Chebyshev numbers C_v(n) for v=10/9: a(n) is the smallest number such that if x>=a(n), then theta(x)-theta(9*x/10)>=n*log(x), where theta(x)=sum_{prime p<=x}log p.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A220475 Chebyshev numbers C_v(n) for v=15/14: a(n) is the smallest number such that if x>=a(n), then theta(x)-theta(14*x/15)>=n*log(x), where theta(x)=sum_{prime p<=x} log p.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A220462 Chebyshev numbers C_v(n) for v=3/2: a(n) is the smallest number such that if x>=a(n), then theta(x)-theta(2x/3)>=nlog(x), where theta(x) = sum_{prime p<=x} log p.

A220463 Chebyshev numbers C_v(n) for v=1.2: a(n) is the smallest number such that if x>=a(n), then theta(x)-theta(5x/6)>=nlog(x), where theta(x)=sum_{prime p<=x}log p.

A220474 Chebyshev numbers C_v(n) for v=10/9: a(n) is the smallest number such that if x>=a(n), then theta(x)-theta(9x/10)>=nlog(x), where theta(x)=sum_{prime p<=x}log p.

A220475 Chebyshev numbers C_v(n) for v=15/14: a(n) is the smallest number such that if x>=a(n), then theta(x)-theta(14x/15)>=nlog(x), where theta(x)=sum_{prime p<=x} log p.