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

1, 3, 8, 14, 20, 36, 48, 66, 70, 96, 126, 132, 156, 240, 252, 300, 336, 450, 480, 540, 660, 690, 714, 870, 900, 1080, 1320, 1470, 1530, 1710, 1950, 2340, 2940, 2970, 3360, 3780, 4200, 4830, 5040, 5610, 5670, 5880, 6270, 7140, 7350, 7410, 8400, 9660, 9870

Offset

1,2

Comments

Conjecture: If n is in the sequence, then the sequence contains an infinite number of multiples of n.

Conjecture: Except for 1 and 3, all members of the sequence are even. If n is odd, it cannot be squarefree.

Conjecture: There does not exist N such that for all n > N, a(n) is divisible by 30.

A036912 gives the values of the phi function at these n.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Example

phi(1)=1, and for n>=1, phi(n)>=1.
phi(3)=2, and for n>=3, phi(n)>=2.
phi(8)=4, and for n>=8, phi(n)>=4.
phi(14)=6, and for n>=14, phi(n)>=6.

Mathematica

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 *)

Prog

(JavaScript)
p = new Array();
p[0] = NaN;
p[1] = 2;
p[2] = 3;
mj = 2;
for (k = 3; k < 50000; k += 2) makeprimes(k);
function makeprimes(i) {
for (j = 2; j <= mj; j++)
if (i%p[j] == 0) return false;
p[++mj] = i;
return true;
}
function primeFactorize(n) {
var pf = new Array(), pc, pfc;
pf[0] = new Array();
pf[1] = new Array();
pc = 1;
pfc = -1;
while (n != 1) {
if (n%p[pc] == 0) {pfc++; pf[0][pfc] = p[pc]; pf[1][pfc] = 0; }
while (n%p[pc] == 0) {n /= p[pc]; pf[1][pfc]++; }
pc++;
}
return pf;
}
function phi(n) {
var f, i, v;
v = 1;
f = primeFactorize(n);
for (i = 0; i < f[0].length; i++) v *= Math.pow(f[0][i], f[1][i] - 1)*(f[0][i] - 1);
return v;
}
function isMin(arr, ik, k) {
var i, im;
im = true;
for (i = ik; i < arr.length; i++) if (arr[i] < k) {im = false; break; }
return im;
}
phiV = new Array();
for (k = 1; k < 50000; k++) phiV[k] = phi(k);
cm = 1;
for (n = 1; n < 3000; n++) if (phiV[n] > cm && isMin(phiV, n, phiV[n])) {cm = phiV[n]; document.write(n + ", "); }

Crossrefs

Cf. A000010, A036912, A057635, A014197.

Sequence in context: A299647 A063617 A062550 * A333962 A366087 A022947

Adjacent sequences: A219927 A219928 A219929 * A219931 A219932 A219933

Keyword

nonn

Author

Jon Perry, Dec 01 2012

Status

approved