A060769 -id:A060769 - cp's OEIS frontend

A060770 "Early" primes: upper ends of prime gaps that are smaller than the prime number theorem predicts.

3, 5, 7, 11, 13, 17, 19, 23, 31, 41, 43, 47, 61, 71, 73, 83, 101, 103, 107, 109, 113, 131, 137, 139, 151, 157, 163, 167, 173, 179, 181, 193, 197, 199, 227, 229, 233, 239, 241, 257, 263, 269, 271, 277, 281, 283, 311

Offset: 1

Ulrich Schimke (ulrschimke(AT)aol.com)

nonn

			11 is in the sequence because 11/log(11) - 7/log(7) = 0.990... < 1.

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

Apart from 2 this is the complementary prime set to A060769.

{ default(realprecision, 100); n=0; s=2/log(2); forprime (p=3, 12583, if ((r=p/log(p)) - s < 1, write("b060770.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.

A060771 Upper ends of record prime gaps under consideration of the prime number theorem.

Original entry on oeis.org

3, 5, 7, 11, 29, 97, 127, 541, 907, 1151, 1361, 15727, 19661, 31469, 156007, 360749, 370373, 1357333, 2010881, 17051887, 20831533, 47326913, 191913031, 436273291, 2300942869, 3842611109, 4302407713, 10726905041, 22367085353, 25056082543

Offset: 1

Ulrich Schimke (ulrschimke(AT)aol.com)

nonn

Every element > 7 must be in A000101 too (consider the derivatives of x/log(x) to prove this), but not conversely. The sequence is infinite since lim sup (length of n-th prime gap/log(n-th prime)) is infinite, proved by Westzynthius, see Ribenboim.

			541 is okay since 541/log(541) - 523/log(523) = 2.4108.. was not reached by smaller primes

P. Ribenboim, The Book of Prime Number Records, Chapter about prime gaps.
E. Westzynthius, Über die Verteilung der Zahlen, die zu den n ersten Primzahlen teilerfremd sind Comm. Phys. Math. Helsingfors 25, 1931.

Cf. A060769, A000101.

A prime p belongs to the sequence iff p/log(p) - q/log(q) attains a new high, 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 *)

A382051 Primes prime(k) such that klog(k)/prime(k) < (k-1)log(k-1)/prime(k-1).

Original entry on oeis.org

11, 17, 23, 29, 37, 53, 59, 67, 79, 89, 97, 127, 137, 149, 157, 163, 173, 179, 191, 211, 223, 239, 251, 257, 263, 269, 277, 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, 787, 797

Offset: 1

Alain Rocchelli, Mar 13 2025

nonn

a(n) ~ prime(round(n*e)) as n tends to infinity, where e is Euler's number.

			11 is a term because 5*log(5)/11 < 4*log(4)/7 and 11 is the 5th prime following 7.
17 is a term because 7*log(7)/17 < 6*log(6)/13 and 17 is the 7th prime following 13.

A subsequence is A060769.

Cf. A382052, A060769, A001113, A068985.

Select[Prime[Range[2,139]],PrimePi[#]*Log[PrimePi[#]]/#<(PrimePi[#]-1)*Log[PrimePi[#]-1]/NextPrime[#,-1]&] (* James C. McMahon, Apr 08 2025 *)

my(N=1); forprime(P=3, 800, my(Q=precprime(P-1), AR0=N*log(N)/Q, AR=(N+1)*log(N+1)/P); N++; if(AR

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

cp's OEIS Frontend

A060770 "Early" primes: upper ends of prime gaps that are smaller than the prime number theorem predicts.

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A060771 Upper ends of record prime gaps under consideration of the prime number theorem.

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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)).

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A382051 Primes prime(k) such that klog(k)/prime(k) < (k-1)log(k-1)/prime(k-1).

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cp's OEIS Frontend

A060770 "Early" primes: upper ends of prime gaps that are smaller than the prime number theorem predicts.

Views

Author

Keywords

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Links

Crossrefs

Programs

PARI

Formula

A060771 Upper ends of record prime gaps under consideration of the prime number theorem.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

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

A382051 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

A382051 Primes prime(k) such that klog(k)/prime(k) < (k-1)log(k-1)/prime(k-1).