A198696 -id:A198696 - cp's OEIS frontend

A066485 Numbers n such that f(n) is a strict local extremum for the prime gaps function f(n) = prime(n+1)-prime(n), where prime(n) denotes the n-th prime; i.e., either f(n)>f(n-1) and f(n)>f(n+1) or f(n)

4, 5, 6, 7, 9, 10, 11, 13, 17, 18, 20, 21, 22, 24, 26, 27, 28, 30, 31, 32, 33, 34, 35, 38, 41, 42, 43, 44, 45, 49, 51, 52, 53, 57, 58, 60, 62, 64, 66, 67, 68, 69, 72, 75, 77, 78, 80, 81, 82, 83, 84, 85, 87, 89, 91, 93, 94, 95, 97, 98, 99, 100, 101, 104, 106, 109, 113, 114

Offset: 1

Joseph L. Pe, Jan 02 2002

nonn

Call a finite subsequence of consecutive terms of a(n) a "zigzag" if it consists of consecutive integers; for example, 30, 31, 32, 33, 34, 35 is a zigzag. Are there zigzags of arbitrary length? (Cf. A066918.)

			4 is a term since f(4) is a local maximum: f(3)=2, f(4)=4, f(5)=2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A066918, A001223.

Cf. A198696 (local maxima), A196174 (local minima).

Primes:= select(isprime,[2,seq(2*i+1,i=1..10^3)]):
G:= Primes[2..-1] - Primes[1..-2]:
select(n -> G[n] > max(G[n-1],G[n+1]) or G[n] < min(G[n-1],G[n+1]), [$2..nops(G)-1]):
# Robert Israel, Sep 20 2015

f[n_] := Prime[n+1]-Prime[n]; Select[Range[200], (f[ # ]-f[ #-1])(f[ # ]-f[ #+1])>0&]

f(n) = prime(n+1)-prime(n);
isok(n) = if (n>2, my(x=f(n), y=f(n-1), z=f(n+1)); ((x>y) && (x>z)) || ((xMichel Marcus, Mar 26 2020

Edited by Dean Hickerson, Jun 26 2002

A198697 Values of local maxima in differences of primes, A001223.

Original entry on oeis.org

4, 4, 6, 6, 6, 6, 8, 4, 14, 6, 10, 10, 4, 6, 10, 6, 14, 14, 10, 8, 8, 10, 10, 6, 8, 12, 8, 12, 18, 10, 10, 12, 12, 10, 10, 8, 8, 14, 12, 10, 4, 14, 14, 20, 10, 14, 8, 12, 6, 10, 10, 10, 18, 8, 22, 10, 10, 12, 12, 18, 6, 6, 12, 34, 18, 14, 8, 12, 4, 12, 8, 8

Offset: 1

Zak Seidov, Oct 29 2011

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Zak Seidov, A001223: maxima of 2nd order

Cf. A198696 (positions of local maxima in A001223).
Cf. A196175 (positions of local minima in A001223).
Cf. A001223 (first differences of primes).

P:= select(isprime,[2,seq(i,i=3..1000,2)]):
DP:= P[2..-1] - P[1..-2]:
J:= select(t -> DP[t] > DP[t-1] and DP[t] > DP[t+1], [$2..nops(DP)-1]):
DP[J]; # Robert Israel, Mar 07 2024

nn = 1001; t = Differences[Prime[Range[nn]]]; t2 = {}; Do[If[t[[n - 1]] < t[[n]] && t[[n]] > t[[n + 1]], AppendTo[t2, {n, t[[n]]}]], {n, 2, nn - 2}]; Transpose[t2][[2]] (* T. D. Noe, Dec 27 2011 *)
Select[Partition[Differences[Prime[Range[400]]],3,1],#[[1]]<#[[2]]>#[[3]]&][[;;,2]] (* Harvey P. Dale, Mar 07 2023 *)

A385466 Primes that are at the end of the local maxima in the sequence of consecutive prime gaps.

Original entry on oeis.org

11, 17, 29, 37, 67, 79, 97, 107, 127, 137, 149, 191, 197, 239, 251, 277, 307, 331, 347, 367, 397, 419, 431, 439, 457, 479, 499, 521, 541, 557, 587, 631, 673, 701, 719, 751, 769, 787, 809, 821, 827, 853, 877, 907, 929, 967, 991, 1009, 1019, 1031, 1049, 1061, 1087

Offset: 1

Emirhan Üçok, Jun 29 2025

This sequence lists the larger prime in each consecutive prime pair where the difference is a local maximum in the sequence of prime gaps.

A local maximum occurs when p(n)-p(n-1) < p(n+1)-p(n) > p(n+2)-p(n+1) where p(n) is the n-th prime.

			The primes 7 and 11 differ by 4, which is larger than the previous gap (2) and the next gap (2). So 11 is in the sequence.

Cf. A001223 (prime gaps), A198696.

Module[{primes = Prime[Range[1, 200]], diffs, res = {}}, diffs = Differences[primes];
Do[If[diffs[[i]] > diffs[[i - 1]] && diffs[[i]] > diffs[[i + 1]],
AppendTo[res, primes[[i + 1]]]], {i, 2, Length[diffs] - 1}]; res]

from sympy import primerange
primes = list(primerange(2, 2000))
diffs = [primes[i+1] - primes[i] for i in range(len(primes)-1)]
local_max_asals = []
for i in range(1, len(diffs)-1):
    if diffs[i] > diffs[i-1] and diffs[i] > diffs[i+1]:
        local_max_asals.append(primes[i+1])
print(local_max_asals[:70])

a(n) = prime(A198696(n)+1). - Michel Marcus, Jul 01 2025

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A066485 Numbers n such that f(n) is a strict local extremum for the prime gaps function f(n) = prime(n+1)-prime(n), where prime(n) denotes the n-th prime; i.e., either f(n)>f(n-1) and f(n)>f(n+1) or f(n)

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A198697 Values of local maxima in differences of primes, A001223.

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Author

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Maple

Mathematica

A385466 Primes that are at the end of the local maxima in the sequence of consecutive prime gaps.

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