A124661 -id:A124661 - cp's OEIS frontend

A385503 Popular primes.

2, 3, 5, 7, 13, 19, 23, 31, 43, 47, 73, 83, 109, 113, 199, 283, 467, 661, 773, 887, 1109, 1129, 1327, 1627, 2143, 2399, 2477, 2803, 2861, 2971, 3739, 3931, 3947, 4297

Offset: 1

Peter Munn, Jul 01 2025

McNew says that a prime p is "popular" on an interval [2, k] if no prime occurs more frequently than p as the greatest prime factor (gpf, A006530) of the integers in that interval. - N. J. A. Sloane, Jul 25 2017

Does there exist two popular primes p < q such that q gets popular earlier than p, i.e., such that q is popular (for the first time) on [2,k] but p is not popular on [2,j] for any j < k? - Pontus von Brömssen, Jul 02 2025

Nathan McNew, Popular values of the largest prime divisor function, arXiv:1504.05985 [math.NT], April 2015.
Nathan McNew, Popular values of the largest prime divisor function (corrected version), page 16, November 2015.
Nathan McNew, The Most Frequent Values of the Largest Prime Divisor Function, Exper. Math., 2017, Vol. 26, No. 2, 210-224.

Cf. A006530, A124661, A246033.

A127925 Primes p such that 2p < prime(k-i) + prime(k+i) for i=1..k-1, where p=prime(k).

Original entry on oeis.org

3, 7, 19, 23, 43, 47, 73, 109, 113, 199, 283, 293, 313, 317, 463, 467, 503, 509, 523, 619, 661, 683, 691, 887, 1063, 1069, 1109, 1129, 1303, 1307, 1321, 1327, 1613, 1621, 1627, 1637, 1669, 1789, 2143, 2161, 2383, 2393, 2399, 2477, 2731, 2753, 2803, 2861, 2971

Offset: 1

T. D. Noe, Feb 06 2007

nonn

One of several sets of "good primes" in section A14 of Guy.

McNew calls these numbers "midpoint convex primes". - Peter Munn, Jul 04 2025

R. K. Guy, Unsolved Problems in Number Theory, 3rd ed. Springer, 2004.

T. D. Noe, Table of n, a(n) for n=1..10001
Nathan McNew, Popular values of the largest prime divisor function (corrected version), page 16, November 2015.

Cf. A028388.
A246033 is a subset.
Subset of A124661, A178954.

t={}; n=1; While[Length[t]<100, n++; p=Prime[n]; i=1; While[i

A319126 Convex hull primes, that is, prime numbers corresponding to the convex hull of PrimePi, the prime counting function.

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 23, 31, 43, 47, 73, 113, 199, 283, 467, 661, 887, 1129, 1327, 1627, 2803, 3947, 4297, 5881, 6379, 7043, 9949, 10343, 13187, 15823, 18461, 24137, 33647, 34763, 37663, 42863, 43067, 59753, 59797, 82619, 96017, 102679, 129643, 130699, 142237

Offset: 1

Jean-François Alcover, Sep 11 2018

nonn

"Convex hull of PrimePi" is a short wording for "the upper convex hull of the points {p, PrimePi(p)} for p >= 2".

			Prime 83 is not member because there exist two primes from the convex hull, namely 47 and 113, such that (PrimePi(83) - PrimePi(47))/(83 - 47) < (PrimePi(113) - PrimePi(83))/(113 - 83).

Wikipedia, Convex hull

Cf. A000720, A124661, A167844, A246033 (a subsequence).

terms = 42;
pMax = 110000;
a[1] = 2;
a[n_] := a[n] = Module[{}, For[slopeMax = 0; p1 = NextPrime[a[n-1]], p1 <= pMax, p1 = NextPrime[p1], slope = (PrimePi[p1] - PrimePi[a[n-1]])/(p1 - a[n-1]); If[slope > slopeMax, slopeMax = slope; p1Max = p1]]; p1Max];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 42}]

lista(nn) = my(c, m, p=2, r, s, t=1); print1(p); for(n=2, nn, c=t; m=0; forprime(q=p+1, oo, c++; if(m0&&sJinyuan Wang, Feb 25 2025

More terms from Jinyuan Wang, Feb 25 2025

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

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A385503 Popular primes.

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A127925 Primes p such that 2p < prime(k-i) + prime(k+i) for i=1..k-1, where p=prime(k).

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A319126 Convex hull primes, that is, prime numbers corresponding to the convex hull of PrimePi, the prime counting function.

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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).

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