A063948 Unitary untouchable numbers: us(x) = n has no solution where us(x) (A063919) is the sum of the proper unitary divisors of x.

2, 3, 4, 5, 7, 374, 702, 758, 998, 1542, 1598, 1778, 1808, 1830, 1974, 2378, 2430, 2910, 3164, 3182, 3188, 3216, 3506, 3540, 3666, 3698, 3818, 3846, 3986, 4196, 4230, 4574, 4718, 4782, 5126, 5324, 5610, 5738, 5918, 5952, 6002, 6174, 6270, 6404, 6450, 6510

Offset

1,1

Comments

Pomerance & Yang show that this sequence has positive lower density (in fact, greater than 10^-7) and upper density at most 0.40632. - Charles R Greathouse IV, Dec 28 2013

Links

Donovan Johnson, Table of n, a(n) for n = 1..10000

C. Pomerance and H.-S. Yang, On untouchable numbers and related problems, 2012

C. Pomerance and H.-S. Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Mathematics of Computation, to appear c. 2014

Formula

If us(x) = n > 1, then n^2 - 4x >= 0. - Dean Hickerson, Sep 04, 2001.

Mathematica

us[x_] := us[x] = Total[ Select[ Divisors[x], GCD[#, x/#] == 1 &]] - x; us[1] = 1; usQ[n_] := With[{xm = Ceiling[n^2/4]}, Catch[ Do[ If[us[x] == n, Throw[True]]; If[x == xm, Throw[False]], {x, 1, xm}]]]; A063948 = Reap[ Do[ If[ !usQ[n], Print[n]; Sow[n]], {n, 1, 6600}]][[2, 1]] (* Jean-François Alcover, Jun 22 2012 *)

Crossrefs

Cf. A034448, A063919, A064000.

Sequence in context: A285888 A028986 A327324 * A113929 A220394 A082351

Adjacent sequences: A063945 A063946 A063947 * A063949 A063950 A063951

Keyword

nonn,nice

Author

Felice Russo, Sep 04 2001

Extensions

More terms from David W. Wilson, Sep 05 2001

Status

approved