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A385504 Binomially timely primes: primes prime(k) that do not arrive late in comparison with the binomially weighted average of prime(1) .. prime(2k-1).

%I A385504 #14 Jul 27 2025 00:21:00
%S A385504 2,3,5,7,13,19,23,31,43,47,53,61,73,79,83,89,103,107,109,113,139,151,
%T A385504 167,181,193,197,199,211,233,239,241,271,277,281,283,293,313,317,353,
%U A385504 359,383,389,401,443,449,461,463,467,479,487,491,499,503,509,521,523,617
%N A385504 Binomially timely primes: primes prime(k) that do not arrive late in comparison with the binomially weighted average of prime(1) .. prime(2k-1).
%C A385504 Primes prime(k) such that prime(k) <= A007443(2k-1)/2^(2k-2), where prime(k) is the k-th prime and A007443 is the binomial transform of primes.
%C A385504 Though the average uses all primes from 2 to prime(2k-1), their influence is substantially weighted towards the primes nearer to prime(k).
%C A385504 Some previously studied sets of primes that depend on each prime's relationship with a broad neighborhood of primes, e.g., convex hull primes (A319126) and A124661, can be shown to be subsets of these timely primes, and some other such sets, e.g., popular primes (A385503), look likely to be shown to be subsets too.
%C A385504 Comments about density within the primes: (Start)
%C A385504 The progressive decrease in density of the primes means this weighted average we are using might be seen as slightly biased so that primes that are "only approximately on time" qualify for the sequence. Nevertheless, this bias in the average seems to be significantly less than 0.5, slowly decreasing with index, and the author expects an analytically derivable asymptote (for the bias) of about 0.25. See also the comments in A302334.
%C A385504 The early race behavior (timely primes v. their complement within the primes) looks like races where the chosen subset's relative asymptotic density is 0.5 and where this subset is ahead except for occasional relatively short excursions where the complement takes over. Here, timely primes are ahead for more than 80% of the indices up to the 500th prime; they then lead continuously up to the 10000th prime, where their lead has fallen below 50 after a peak greater than 200. See the graph in the links. (End)
%H A385504 Peter Munn, <a href="/A385504/b385504.txt">Table of n, a(n) for n = 1..2000</a>
%H A385504 Peter Munn, <a href="/A385504/a385504.txt">PARI program</a>
%H A385504 Peter Munn, <a href="/A385504/a385504.png">Graph of race against complement within primes</a>
%F A385504 {a(n) : n >= 1} = {prime(k) : k >= 1 and prime(k) <= A302334(k)}.
%e A385504 The binomially weighted averages can be computed by taking progressive averages as shown in the table below:
%e A385504    n   prime |<- progressive averages ... ->
%e A385504   -------------------------------------------
%e A385504    1:   _2_                              the _underlined_ values are the averaged primes
%e A385504               5/2
%e A385504    2:    3         _13/4_                   <-- 13/4 is thus the 2nd averaged prime
%e A385504                4            33/8
%e A385504    3:    5           5            _83/16_       <-- 83/16 is thus the 3rd averaged prime
%e A385504                6            25/4  ...
%e A385504    4:    7          15/2   ...              <-- 15/2 is the average of 6 and 9
%e A385504                9  ...
%e A385504    5:   11  ...
%e A385504   ...
%e A385504 3 is less than 13/4, so 3 is in the sequence.
%e A385504 5 is less than 83/16, so 5 is in the sequence.
%e A385504 If we continue the average table above, we find the 5th averaged prime is 10 + 147/256, and the 5th prime, 11, is greater than this, so 11 is not in the sequence.
%o A385504 (PARI) \\ See Links
%Y A385504 See the comments for the relationship with A007443.
%Y A385504 See the formula section for the relationship with A302334.
%Y A385504 Cf. A060770, A385503.
%Y A385504 A124661, A319126 are subsets.
%K A385504 nonn
%O A385504 1,1
%A A385504 _Peter Munn_, Jul 11 2025