OFFSET
1,1
LINKS
Donovan Johnson, Table of n, a(n) for n = 1..10000
Carl Pomerance and Hee-Sung Yang, On untouchable numbers and related problems, 2012.
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Mathematics of Computation, Vol. 83, No. 288 (2014), pp. 1903-1913; alternative link.
FORMULA
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.
MATHEMATICA
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 *)
PROG
(PARI) usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]); }
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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Labos Elemer and Felice Russo, Sep 05 2001
EXTENSIONS
Edited by N. J. A. Sloane, May 04 2007
STATUS
approved