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 A190661 Least number a(n) such that there are at least n primes in the range (T(k-1), T(k)] for all k >= a(n), where T(k) is the k-th triangular number. 8

%I

%S 1,7,16,33,52,66,79,72,109,93,121,119,130,153,169,194,180,222,235,275,

%T 294,267,256,296,329,339,333,420,383,373,372,454,396,443,449,504,463,

%U 574,559,512,592,602,596,541,652,585,683,656,687,689,708

%N Least number a(n) such that there are at least n primes in the range (T(k-1), T(k)] for all k >= a(n), where T(k) is the k-th triangular number.

%C All values and even whether the sequence is well defined are conjectural.

%C a(n) is the conjectured index of the last occurrence of n in A066888.

%C It is conjectured that for every n >= 0, a(n) > n.

%C With R_n the n-th Ramanujan prime (A104272), it is conjectured that for every n >= 0, (1/2) R_n <= a(n) < (20/13) R_n. These bounds have been verified for all n up to 8000. For most n <= 8000, we have a(n) > R_n, with exceptions listed in A190881.

%H T. D. Noe, <a href="/A190661/b190661.txt">Table of n, a(n) for n = 0..8000</a>

%e Because it appears that A066888(7) = 1 is the last 1 of that sequence, a(1) = 7.

%Y Cf. A066888, A000217, A000040, A088634, A104272, A190881.

%K nonn

%O 0,2

%A _John W. Nicholson_, May 18 2011

%E Edited by _T. D. Noe_, May 19 2011

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Last modified January 20 07:59 EST 2020. Contains 331081 sequences. (Running on oeis4.)