A066918 -id:A066918 - 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

A066923 Let f(x) = phi(x) + sigma(x); a(n) = least k such that at k begins a maximal run of length n of consecutive strict local extrema of f, or 0 if no such k exists.

Original entry on oeis.org

118, 12, 443, 190, 40, 16, 5847, 180, 108, 48, 1427, 670510, 2388, 228, 407, 1577, 424, 2500, 2500383, 22848, 4853, 1240, 323975, 0, 10668, 588, 10727, 45677, 18713, 1903672, 0, 0, 119028, 18880, 391659, 0, 883428, 480036, 1635467, 896933, 50380

Offset: 1

Joseph L. Pe, Jan 23 2002

nonn

A084622 gives the strict local extrema for f. A run of consecutive strict local extrema of a function is sometimes called a zigzag, cf. A066485. A066918 is an analog of the present sequence for the prime gaps function.

The zero terms a(24), a(31), a(32), a(36) are preliminary since only values of f(n) for n up to 6000000 were taken into account. Further nonzero terms are a(45) = 1413696, a(46) = 185195, a(49) = 4961856, a(50) = 2370036.

			f(10) = 22, f(11) = 22, f(12) = 32, f(13) = 26, f(14) = 30, f(15) = 32. A run of length 2 begins at f(12) = 32 because f(12) = 32 is a local maximum and f(13) = 26 is local minimum.
This is a maximal run, since neither f(11) = 22 nor f(14) = 30 are local extrema of f. Also, a maximal run of length 2 first occurs at f(12) = 32. Therefore a(2) = 12.

Cf. A066485, A066918, A084622.

f[ n_ ] := EulerPhi[ n ] + DivisorSigma[ 1, n ]; e[ n_ ] := (f[ n - 1 ] < f[ n ] && f[ n + 1 ] < f[ n ]) || (f[ n - 1 ] > f[ n ] && f[ n + 1 ] > f[ n ]); z[ n_, k_ ] := Module[ {r = True, i = 0}, While[ i <= k && r == True, If[ e[ n + i ], r = False ]; i++ ]; r ]; z2[ n_, k_ ] := z[ n, k ] && ! e[ n + k + 1 ] && ! e[ n - 1 ]; k[ n_ ] := Module[ {i = 2, r = False}, While[ r == False && i < 10^6, If[ z2[ i, n ], r = True; Print[ i ] ]; i++ ]; If[ r == False, Print[ "0" ] ] ]; Table[ {i, k[ i ]}, {i, 0, 17} ]

f(x)=eulerphi(x)+sigma(x)
{locext(n)=local(a,b,c); a=if(n<2,0,f(n-1)); b=f(n); c=f(n+1); if(ac,1,if(a>b&&b

Edited, corrected (a(12)) and extended (a(19) ff.) by Klaus Brockhaus, Jun 01 2003

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)

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