A383216 -id:A383216 - cp's OEIS frontend

A383215 Primes p preceded and followed by gaps whose difference (absolute value) is greater than log(p).

7, 29, 31, 113, 127, 139, 149, 181, 191, 199, 223, 241, 283, 307, 317, 331, 347, 419, 421, 431, 467, 521, 523, 541, 619, 641, 661, 673, 773, 809, 811, 821, 829, 853, 863, 877, 887, 907, 953, 967, 1009, 1021, 1031, 1049, 1051, 1061, 1069, 1087, 1129, 1151, 1153, 1213, 1259, 1277

Offset: 1

Alain Rocchelli, Apr 19 2025

nonn

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

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

			7 is a term because abs(5-2*7+11)=2 and log(7)=1.9459.
29 is a term because abs(23-2*29+31)=4 and log(29)=3.3673.

Cf. A036263, A383216, A068985.

Select[Prime[Range[2,206]],Abs[NextPrime[#,-1]-2#+NextPrime[#]]>Log[#]&] (* James C. McMahon, Apr 29 2025 *)

forprime(P=3, 1300, my(M=P-precprime(P-1), Q=nextprime(P+1)-P, AR1=min(M,Q), AR2=max(M,Q), AR0=log(P)); if(AR2-AR1>AR0, print1(P,", ")));

Limit_{n->oo} n / PrimePi(a(n)) = 1/e (A068985).

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.

cp's OEIS Frontend

A383215 Primes p preceded and followed by gaps whose difference (absolute value) is greater than log(p).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI