1,1

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 analogue 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.

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} ]

(PARI) 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(a<b&&b>c, 1, if(a>b&&b<c, -1, 0))}

{m=6000000; u=50; v=vector(u); n=1; while(n<m, if((a=locext(n))==0, n++, s=n; n++; c=1; while((b=locext(n))==-a, a=b; c++; n++); if(c<=u&&v[c]==0, v[c]=s))); v}

Cf. A066485, A066918, A084622.

Sequence in context: A027511 A079196 A113798 this_sequence A039556 A095627 A066734

Adjacent sequences: A066920 A066921 A066922 this_sequence A066924 A066925 A066926

nonn

Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Jan 23 2002

Edited, corrected (a(12)) and extended (a(19) ff.) by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jun 01 2003