cp's OEIS Frontend

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.

%I A066923 #11 Jan 03 2017 02:26:23
%S A066923 118,12,443,190,40,16,5847,180,108,48,1427,670510,2388,228,407,1577,
%T A066923 424,2500,2500383,22848,4853,1240,323975,0,10668,588,10727,45677,
%U A066923 18713,1903672,0,0,119028,18880,391659,0,883428,480036,1635467,896933,50380
%N 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.
%C A066923 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.
%C A066923 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.
%e A066923 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.
%e A066923 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.
%t A066923 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} ]
%o A066923 (PARI) f(x)=eulerphi(x)+sigma(x)
%o A066923 {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))}
%o A066923 {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}
%Y A066923 Cf. A066485, A066918, A084622.
%K A066923 nonn
%O A066923 1,1
%A A066923 _Joseph L. Pe_, Jan 23 2002
%E A066923 Edited, corrected (a(12)) and extended (a(19) ff.) by _Klaus Brockhaus_, Jun 01 2003