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A063948
Unitary untouchable numbers: us(x) = n has no solution where us(x) (A063919) is the sum of the proper unitary divisors of x.
14
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
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
KEYWORD
nonn,nice
AUTHOR
Felice Russo, Sep 04 2001
EXTENSIONS
More terms from David W. Wilson, Sep 05 2001
STATUS
approved