A302334 -id:A302334 - cp's OEIS frontend

A356627 Primes whose powers appear in A332979.

2, 3, 5, 7, 11, 17, 29, 37, 41, 59, 67, 71, 97, 127, 149, 191, 223, 269, 307, 347, 419, 431, 557, 563, 569, 587, 593, 599, 641, 727, 809, 937, 967, 1009, 1213, 1277, 1423, 1861, 1973, 2237, 2267, 2657, 3163, 3299, 3449, 3457, 3527, 3907, 4001, 4211, 4441, 4637

Offset: 1

Michael De Vlieger, Sep 27 2022

nonn

Maxima of row n > 0 of A005940, A182944, and A182945 are powers of these primes.

Indices k of primes, A000040(k), listed here show an interesting correlation with the function f(k) = A000040(k) - A302334(k). - Peter Munn, Sep 29 2022

			5 | A332979(5..7), thus 5 is in the sequence.
7 | A332979(8), thus 7 is in the sequence.
13 does not divide any term in A332979, so it is not a term in this sequence.

Michael De Vlieger, Table of n, a(n) for n = 1..100
Michael De Vlieger, Plot A332979(n) = p^e at (e, pi(p)) for n = 1..360. The primes p are labeled in bold and appear in this sequence. The least and greatest exponents of p^e are labeled in italic.

Cf. A000040, A000961, A005940, A180944, A180945, A302334, A332979.

Prime@ Union@ Table[MaximalBy[Table[{k, n - k}, {k, n}], Prime[#1]^#2 & @@ # &][[1, 1]], {n, 2^10}]
(* or use concise file in A332979 *)
Prime /@ Union@ Rest@ Map[ToExpression@ StringTrim[#, "p"] & @@ StringSplit[#, "^"] &, Import["https://oeis.org/A332979/a332979.txt", "Data"][[All, -1]]]

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).

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 23, 31, 43, 47, 53, 61, 73, 79, 83, 89, 103, 107, 109, 113, 139, 151, 167, 181, 193, 197, 199, 211, 233, 239, 241, 271, 277, 281, 283, 293, 313, 317, 353, 359, 383, 389, 401, 443, 449, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 617

Offset: 1

Peter Munn, Jul 11 2025

nonn

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.
Though the average uses all primes from 2 to prime(2k-1), their influence is substantially weighted towards the primes nearer to prime(k).
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.
Comments about density within the primes: (Start)
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.
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)

			The binomially weighted averages can be computed by taking progressive averages as shown in the table below:
   n   prime |<- progressive averages ... ->
  -------------------------------------------
   1:   _2_                              the _underlined_ values are the averaged primes
              5/2
   2:    3         _13/4_                   <-- 13/4 is thus the 2nd averaged prime
               4            33/8
   3:    5           5            _83/16_       <-- 83/16 is thus the 3rd averaged prime
               6            25/4  ...
   4:    7          15/2   ...              <-- 15/2 is the average of 6 and 9
               9  ...
   5:   11  ...
  ...
3 is less than 13/4, so 3 is in the sequence.
5 is less than 83/16, so 5 is in the sequence.
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.

Peter Munn, Table of n, a(n) for n = 1..2000
Peter Munn, PARI program
Peter Munn, Graph of race against complement within primes

See the comments for the relationship with A007443.
See the formula section for the relationship with A302334.
Cf. A060770, A385503.
A124661, A319126 are subsets.

PARI
```
\\ See Links
```

{a(n) : n >= 1} = {prime(k) : k >= 1 and prime(k) <= A302334(k)}.

A072388 a(0) = 1; for n > 0, a(n) = floor((prime(n+1) + prime(n) + a(n-1))/3).

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 13, 16, 19, 23, 27, 31, 36, 40, 43, 47, 53, 57, 61, 66, 70, 74, 78, 83, 89, 95, 99, 103, 106, 109, 116, 124, 130, 135, 141, 147, 151, 157, 162, 167, 173, 177, 183, 189, 193, 196, 202, 212, 220, 225, 229, 233, 237, 243, 250, 256, 262, 267, 271, 276

Offset: 0

Roger L. Bagula, Jul 20 2002

nonn

A smoothing applied to the primes as a data set. The method is a hybrid of exponential smoothing and simple moving average (see Wikipedia link). - Peter Munn, Mar 26 2019

			a(1) = floor((3+2+1)/3) = 2;
a(2) = floor((5+3+2)/3) = 3.

Wikipedia, Smoothing

Cf. A000040, A024675, A302334.

lista(nn) = {my(a = 1); print1(a, ", "); for (n=1, nn, a = (prime(n+1) + prime(n) + a)\3; print1(a, ", "););} \\ Michel Marcus, Mar 20 2019

Edited by Dean Hickerson, Oct 29 2002

cp's OEIS Frontend

A356627 Primes whose powers appear in A332979.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A072388 a(0) = 1; for n > 0, a(n) = floor((prime(n+1) + prime(n) + a(n-1))/3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions