A060770 -id:A060770 - cp's OEIS frontend

A060769 "Overdue" primes: upper ends of prime gaps that are larger than the prime number theorem predicts.

29, 37, 53, 59, 67, 79, 89, 97, 127, 149, 191, 211, 223, 251, 293, 307, 331, 347, 367, 397, 409, 419, 431, 457, 479, 487, 499, 521, 541, 557, 587, 631, 641, 673, 691, 701, 709, 719, 727, 751, 769

Offset: 1

Ulrich Schimke (ulrschimke(AT)aol.com)

nonn

			79 is in the sequence because 79/log(79) - 73/log(73) = 1.065... > 1.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A060770, A060771.

Apart from 2 this is the complementary prime set to A060770.

{ default(realprecision, 100); n=0; s=2/log(2); forprime (p=3, 26203, if ((r=p/log(p)) - s > 1, write("b060769.txt", n++, " ", p); ); s=r; ) } \\ Harry J. Smith, Jul 11 2009

A prime p belongs to the sequence iff p/log(p) - q/log(q) > 1 where q is the preceding prime.

A330823 a(1) = 1; for n > 1, a(n) = a(n-1) - n if n is prime, otherwise a(n) = a(n-1) + floor(n/(log(n)-1)).

Original entry on oeis.org

1, -1, -4, 6, 1, 8, 1, 8, 15, 22, 11, 19, 6, 14, 22, 31, 14, 23, 4, 14, 24, 34, 11, 22, 33, 44, 55, 67, 38, 50, 19, 31, 44, 57, 70, 83, 46, 60, 74, 88, 47, 62, 19, 34, 50, 66, 19, 35, 51, 68, 85, 102, 49, 67, 85, 103, 121, 139, 80, 99, 38, 57, 77, 97, 117, 137, 70

Offset: 1

Scott R. Shannon, Jan 02 2020

The Prime Number Theorem shows that the probability of a random number not greater than x being prime is approximately 1/log(x), therefore the probability of a number being composite in the same range is approximately (log(x)-1)/log(x). As this sequence subtracts n from the previous term if n is prime, or adds n with a weighting of 1/(log(n)-1) if n is composite, its expected value as n goes to infinity is approximately n*(1/(log(n)-1))*((log(n)-1)/log(n)) - n*(1/log(n)) = 0. We therefore expect that a(n)/n approaches 0 as n goes to infinity.

In the first 2 million terms the sequence changes sign 1900 times, has a maximum positive value of 160213275 at a(1772200), and a maximum negative value of -29535301 at a(1513751). The majority of terms are positive. See the image link below.

Scott R. Shannon, Graph showing a(n) for n = 1 to 2000000. The blue line is the x axis.
Wikipedia, Prime Number Theorem and Prime Counting Function.

Cf. A000040, A057835, A060769, A060770, A050499, A085581.

a[1] = 1; a[n_] := a[n] = a[n - 1] + If[PrimeQ[n], -n, Floor[n/(Log[n] - 1)]]; Array[a, 67] (* Amiram Eldar, Jan 05 2020 *)

A382052 Primes prime(k) such that klog(k)/prime(k) > (k-1)log(k-1)/prime(k-1).

Original entry on oeis.org

3, 5, 7, 13, 19, 31, 41, 43, 47, 61, 71, 73, 83, 101, 103, 107, 109, 113, 131, 139, 151, 167, 181, 193, 197, 199, 227, 229, 233, 241, 271, 281, 283, 311, 313, 317, 337, 349, 353, 359, 373, 379, 383, 389, 401, 421, 433, 439, 443, 449, 461, 463, 467, 491, 503, 509, 523, 547, 563, 569, 571, 577, 593, 599

Offset: 1

Alain Rocchelli, Mar 13 2025

nonn

All terms of this sequence are contained in A060770.

a(n) ~ prime(round(n*e/(e-1))) as n tends to infinity, cf. A185393.

			3 is a term because 2*log(2)/3 > 1*log(1)/2 and 3 is the 2nd prime following 2.
5 is a term because 3*log(3)/5 > 2*log(2)/3 and 5 is the 3rd prime following 3.

Cf. A382051, A060770, A068996, A185393.

Select[Prime[Range[2, 109]],PrimePi[#]*Log[PrimePi[#]]/#>(PrimePi[#]-1)*Log[PrimePi[#]-1]/NextPrime[#,-1]&] (* James C. McMahon, Apr 14 2025 *)

my(N=1); forprime(P=3, 600, my(Q=precprime(P-1), AR0=N*log(N)/Q, AR=(N+1)*log(N+1)/P); N++; if(AR>AR0, print1(P,", ")));

Limit_{n->oo} n / PrimePi(a(n)) = 1-1/e (A068996).

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)}.

cp's OEIS Frontend

A060769 "Overdue" primes: upper ends of prime gaps that are larger than the prime number theorem predicts.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A330823 a(1) = 1; for n > 1, a(n) = a(n-1) - n if n is prime, otherwise a(n) = a(n-1) + floor(n/(log(n)-1)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A382052 Primes prime(k) such that k*log(k)/prime(k) > (k-1)*log(k-1)/prime(k-1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

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

A382052 Primes prime(k) such that klog(k)/prime(k) > (k-1)log(k-1)/prime(k-1).