cp's OEIS Frontend

A064000 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.

%I A064000 #29 Jun 09 2024 09:05:09
%S A064000 2,7,11,13,15,16,19,21,22,23,25,27,29,31,34,35,37,39,41,43,45,46,47,
%T A064000 49,51,52,53,55,57,58,59,61,63,64,66,67,69,71,73,75,76,77,79,81,83,85,
%U A064000 86,87,88,89,91,92,93,94,95,97,99,101,103,105,106,107,109,111,113,115,116
%N A064000 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 A064000 Donovan Johnson, <a href="/A064000/b064000.txt">Table of n, a(n) for n = 1..10000</a>
%H A064000 Carl Pomerance and Hee-Sung Yang, <a href="http://www.math.dartmouth.edu/~carlp/uupaper3.pdf">On untouchable numbers and related problems</a>, 2012.
%H A064000 Carl Pomerance and Hee-Sung Yang, <a href="https://doi.org/10.1090/S0025-5718-2013-02775-5">Variant of a theorem of Erdős on the sum-of-proper-divisors function</a>, Mathematics of Computation, Vol. 83, No. 288 (2014), pp. 1903-1913; <a href="http://www.math.dartmouth.edu/~carlp/uupaper6.pdf">alternative link</a>.
%F A064000 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 A064000 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 *)
%o A064000 (PARI) usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]);}
%o A064000 lista(kmax) = {my(v = vector(kmax), s); for(k = 1, kmax, s = usigma(k); if(s <= kmax, v[s]++)); for(k = 1, kmax, if(v[k] == 0, print1(k, ", ")))}; \\ _Amiram Eldar_, Jun 09 2024
%Y A064000 Cf. A034448, A063948.
%K A064000 easy,nonn
%O A064000 1,1
%A A064000 _Labos Elemer_ and _Felice Russo_, Sep 05 2001
%E A064000 Edited by _N. J. A. Sloane_, May 04 2007