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

Author

Felice Russo, Sep 04 2001

Keywords

• nonn
• nice

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

Crossrefs

Cf. A034448, A063919, A064000.

Programs

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 *)

Formula

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

Extensions

More terms from David W. Wilson, Sep 05 2001