A383215 -id:A383215 - cp's OEIS frontend

A383216 Primes p which are preceded and followed by gaps whose difference is greater than 2*log(p).

113, 127, 523, 887, 907, 1087, 1129, 1151, 1277, 1327, 1361, 1669, 1693, 1931, 1951, 1973, 2203, 2311, 2333, 2477, 2557, 2971, 2999, 3163, 3251, 3299, 3469, 4049, 4297, 4327, 4523, 4547, 4783, 4861, 5119, 5147, 5237, 5351, 5381, 5531, 5557, 5591, 5749, 5779, 5981

Offset: 1

Alain Rocchelli, Apr 19 2025

nonn

Primes prime(k) such that abs(prime(k-1)-2*prime(k)+prime(k+1)) > 2*log(prime(k)), where log is the natural logarithm.

			113 is a term because abs(109-2*113+127)=12 and 2*log(113)=9.4548.
127 is a term because abs(113-2*127+131)=10 and 2*log(127)=9.6884.

Cf. A036263, A383215, A092553.

Select[Prime[Range[2,782]],Abs[NextPrime[#,-1]-2#+NextPrime[#]]>2Log[#]&] (* James C. McMahon, Apr 27 2025 *)

forprime(P=3, 6000, my(M=P-precprime(P-1), Q=nextprime(P+1)-P, AR1=min(M,Q), AR2=max(M,Q), AR0=2*log(P)); if(AR2-AR1>AR0, print1(P,", ")));

Limit_{n->oo} n / PrimePi(a(n)) = 1/e^2 (A092553).

A384603 Primes preceded and followed by gaps whose quotient (value greater or equal to 1) is less than 2.

Original entry on oeis.org

5, 23, 37, 47, 53, 67, 79, 83, 89, 131, 157, 163, 167, 173, 211, 233, 251, 257, 263, 277, 293, 337, 353, 359, 367, 373, 379, 383, 389, 409, 439, 443, 449, 479, 503, 547, 557, 563, 577, 587, 593, 607, 613, 631, 647, 653, 677, 683, 691, 701, 709, 719, 727, 733, 739, 751, 757, 787, 797

Offset: 1

Alain Rocchelli, Jun 04 2025

nonn

Primes prime(k) such that Max(prime(k)-prime(k-1),prime(k+1)-prime(k)) / Min(prime(k)-prime(k-1),prime(k+1)-prime(k)) < 2.

			5 is a term because Max(5-3,7-5)/Min(5-3,7-5) = 2/2 = 1.
23 is a term because Max(23-19,29-23)/Min(23-19,29-23) = 6/4 = 1.5.
37 is a term because Max(37-31,41-37)/Min(37-31,41-37) = 6/4 = 1.5.

Cf. A383215.

forprime(P=3, 1000, my(M=P-precprime(P-1), Q=nextprime(P+1)-P, AR=max(M,Q)/min(M,Q), AR0=2); if(AR

from itertools import islice
from sympy import nextprime
def A384603_gen(): # generator of terms
    p,q,r = 2,3,5
    while True:
        s, t = q-p, r-q
        if s<(t<<1) and t<(s<<1): yield q
        p, q, r = q, r, nextprime(r)
A384603_list = list(islice(A384603_gen(),59)) # Chai Wah Wu, Jun 10 2025

Conjecture: Limit_{n->oo} n / PrimePi(a(n)) = 1/3.

A383591 Smallest prime p where the absolute difference of the gaps to the adjacent primes exceeds n*log(p).

Original entry on oeis.org

7, 113, 1327, 15683, 31397, 31397, 360653, 1349533, 1357333, 17051887, 20831323, 47326913, 436273291, 3842610773, 3842610773, 22367084959, 25056082087, 25056082087, 304599509051, 1346294310749

Offset: 1

Jean-Marc Rebert, May 03 2025

			a(3) = 1327 because we have abs(1321 - 2*1327 + 1361) = 28, which is greater than 3*log(1327) ≈ 21.57, and no smaller prime meets this condition.

Cf. A036263, A383215, A383216.

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A383216 Primes p which are preceded and followed by gaps whose difference is greater than 2*log(p).

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A384603 Primes preceded and followed by gaps whose quotient (value greater or equal to 1) is less than 2.

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Python

Formula

A383591 Smallest prime p where the absolute difference of the gaps to the adjacent primes exceeds n*log(p).

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Author

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Examples

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