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A219930 n such that phi(n) represents a new lower bound for the phi function.

%I A219930 #45 Aug 09 2015 16:03:00
%S A219930 1,3,8,14,20,36,48,66,70,96,126,132,156,240,252,300,336,450,480,540,
%T A219930 660,690,714,870,900,1080,1320,1470,1530,1710,1950,2340,2940,2970,
%U A219930 3360,3780,4200,4830,5040,5610,5670,5880,6270,7140,7350,7410,8400,9660,9870
%N A219930 n such that phi(n) represents a new lower bound for the phi function.
%C A219930 Conjecture: If n is in the sequence, then the sequence contains an infinite number of multiples of n.
%C A219930 Conjecture: Except for 1 and 3, all members of the sequence are even. If n is odd, it cannot be squarefree.
%C A219930 Conjecture: There does not exist N such that for all n > N, a(n) is divisible by 30.
%C A219930 A036912 gives the values of the phi function at these n.
%H A219930 Vincenzo Librandi, <a href="/A219930/b219930.txt">Table of n, a(n) for n = 1..200</a>
%e A219930 phi(1)=1, and for n>=1, phi(n)>=1.
%e A219930 phi(3)=2, and for n>=3, phi(n)>=2.
%e A219930 phi(8)=4, and for n>=8, phi(n)>=4.
%e A219930 phi(14)=6, and for n>=14, phi(n)>=6.
%t A219930 nn = 8!; t = Table[EulerPhi[n], {n, nn}]; min = Infinity; t2 = {}; Do[If[t[[n]] <= min, AppendTo[t2, {n, t[[n]]}]; min = t[[n]]], {n, Length[t], 1, -1}]; t2 = Reverse[t2]; t3 = {}; mx = 0; Do[If[i[[2]] > mx, mx = i[[2]]; AppendTo[t3, i[[1]]]], {i, t2}]; t3 (* _T. D. Noe_, Dec 04 2012 *)
%o A219930 (JavaScript)
%o A219930 p = new Array();
%o A219930 p[0] = NaN;
%o A219930 p[1] = 2;
%o A219930 p[2] = 3;
%o A219930 mj = 2;
%o A219930 for (k = 3; k < 50000; k += 2) makeprimes(k);
%o A219930 function makeprimes(i) {
%o A219930 for (j = 2; j <= mj; j++)
%o A219930 if (i%p[j] == 0) return false;
%o A219930 p[++mj] = i;
%o A219930 return true;
%o A219930 }
%o A219930 function primeFactorize(n) {
%o A219930 var pf = new Array(), pc, pfc;
%o A219930 pf[0] = new Array();
%o A219930 pf[1] = new Array();
%o A219930 pc = 1;
%o A219930 pfc = -1;
%o A219930 while (n != 1) {
%o A219930 if (n%p[pc] == 0) {pfc++; pf[0][pfc] = p[pc]; pf[1][pfc] = 0;}
%o A219930 while (n%p[pc] == 0) {n /= p[pc]; pf[1][pfc]++;}
%o A219930 pc++;
%o A219930 }
%o A219930 return pf;
%o A219930 }
%o A219930 function phi(n) {
%o A219930 var f, i, v;
%o A219930 v = 1;
%o A219930 f = primeFactorize(n);
%o A219930 for (i = 0; i < f[0].length; i++) v *= Math.pow(f[0][i], f[1][i] - 1)*(f[0][i] - 1);
%o A219930 return v;
%o A219930 }
%o A219930 function isMin(arr, ik, k) {
%o A219930 var i, im;
%o A219930 im = true;
%o A219930 for (i = ik; i < arr.length; i++) if (arr[i] < k) {im = false; break;}
%o A219930 return im;
%o A219930 }
%o A219930 phiV = new Array();
%o A219930 for (k = 1; k < 50000; k++) phiV[k] = phi(k);
%o A219930 cm = 1;
%o A219930 for (n = 1; n < 3000; n++) if (phiV[n] > cm && isMin(phiV, n, phiV[n])) {cm = phiV[n]; document.write(n + ", ");}
%Y A219930 Cf. A000010, A036912, A057635, A014197.
%K A219930 nonn
%O A219930 1,2
%A A219930 _Jon Perry_, Dec 01 2012