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Index for the bound for the first k-Ramanujan prime.
3

%I #23 Aug 03 2017 12:42:14

%S 3,5,7,10,12,16,31,35,47,48,63,67,100,218,264,298,328,368,430,463,591,

%T 651,739,758,782,843,891,929,1060,1184,1230,1316,1410,1832,2226,3386,

%U 3645,3794,3796,4523,4613,4755,5009,5950

%N Index for the bound for the first k-Ramanujan prime.

%C The index a(n) is h(n), the prime A277718(n) is p_h(n). If 1 <= n <= 43 and k in [p_{h(n+1)}/p_{h(n+1)-1}, p_{h(n)}/p_{h(n)-1}), then the first k-Ramanujan prime R^{(k)}_1 = p_{h(n)}. Extra terms require improvements of prime numbers in short intervals.

%H Christian Axler and Thomas Leßmann, <a href="http://arxiv.org/abs/1504.05485">An explicit upper bound for the first k-Ramanujan prime</a>, arXiv:1504.05485 [math.NT], 2015.

%H Christian Axler and Thomas Leßmann, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.124.7.642">On the first k-Ramanujan prime</a>, Amer. Math. Monthly, 124 (2017), 642-646.

%e With n = 3, we see p_h(3) = 17, p_h(4) = 29, so that 29/23 <= k < 17/13. If k = 1.3 then R^(1.3)_1 = 17 = p_h(3).

%Y Cf. A277718, A164952, A104272, A290394 (first (1 + 1/n)-Ramanujan prime).

%K nonn

%O 1,1

%A _John W. Nicholson_, Oct 27 2016