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%I #22 Dec 09 2016 06:19:44
%S 2,7,11,13,15,16,19,21,22,23,25,27,29,31,34,35,37,39,41,43,45,46,47,
%T 49,51,52,53,55,57,58,59,61,63,64,66,67,69,71,73,75,76,77,79,81,83,85,
%U 86,87,88,89,91,92,93,94,95,97,99,101,103,105,106,107,109,111,113,115,116
%N Unitary untouchable numbers of second kind: numbers n such that usigma(x) = n has no solution, where usigma(x) (A034448) is the sum of unitary divisors of x.
%H Donovan Johnson, <a href="/A064000/b064000.txt">Table of n, a(n) for n = 1..10000</a>
%H C. Pomerance and H.-S. Yang, <a href="http://www.math.dartmouth.edu/~carlp/uupaper3.pdf">On untouchable numbers and related problems</a>, 2012
%H C. Pomerance and H.-S. Yang, <a href="http://www.math.dartmouth.edu/~carlp/uupaper6.pdf">Variant of a theorem of Erdos on the sum-of-proper-divisors function</a>, 2012
%F Suppose usigma(x)=n. Then by definition usigma(x)=n>1 for n>1. Let x be a prime. Then usigma(x)=x+1 and so n=x+1. For x not prime, of course, x+1<n. So in general x<=n-1.
%t usigma[n_] := Sum[ Boole[GCD[d, n/d] == 1]*d, {d, Divisors[n]}]; untouchableQ[n_] := (r = True; x = 1; While[x <= n, If[usigma[x] == n, r = False; Break[], x++]]; r); Select[Range[120], untouchableQ] (* _Jean-François Alcover_, Jan 03 2013 *)
%Y Cf. A034448, A063948.
%K easy,nonn
%O 1,1
%A _Labos Elemer_ and _Felice Russo_, Sep 05 2001
%E Edited by _N. J. A. Sloane_, May 04 2007
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