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
KEYWORD
nonn
AUTHOR
Jon Perry, Dec 01 2012
STATUS
approved