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A220269
a(n) is the smallest number, such that for all N >= a(n) there are at least n primes between 3*N and 4*N.
5
2, 8, 11, 17, 26, 38, 40, 41, 48, 57, 68, 68, 70, 87, 96, 100, 108, 109, 110, 115, 136, 149, 151, 161, 161, 169, 176, 178, 184, 206, 208, 227, 235, 236, 242, 255, 259, 260, 263, 272, 297, 299, 305, 320, 356, 358, 359, 371, 371, 372, 378, 386, 389, 392, 400
OFFSET
1,1
LINKS
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
FORMULA
a(n) <= ceiling(R_(4/3)(n)/4), where R_v(n) (v>1) are generalized Ramanujan numbers (see Shevelev's link). In particular, for n >= 1, {R_(4/3)(n)}={11, 29, 59, 67, 101, 149, 157, 163, 191, 227, 269, 271, 307, 379, ...}. Moreover, if R_(4/3)(n) == 1 (mod 4), then a(n) = ceiling(R_(4/3)(n)/4).
MATHEMATICA
nn = 60; t = Table[PrimePi[4 n] - PrimePi[3 n], {n, 10*nn}]; Join[{2}, Table[s = Flatten[Position[t, _?(# > n - 1 &)]]; i = Length[s]; While[i > 1 && s[[i]] - s[[i - 1]] == 1, i--]; s[[i]], {n, 2, nn}]] (* Michael B. Porter, after A220268 program by T. D. Noe, Feb 09 2013 *)
CROSSREFS
Sequence in context: A366915 A064105 A129516 * A143189 A056550 A074263
KEYWORD
nonn
STATUS
approved