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.

2, 7, 11, 13, 15, 16, 19, 21, 22, 23, 25, 27, 29, 31, 34, 35, 37, 39, 41, 43, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 63, 64, 66, 67, 69, 71, 73, 75, 76, 77, 79, 81, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 99, 101, 103, 105, 106, 107, 109, 111, 113, 115, 116

Offset: 1

Author

Labos Elemer and Felice Russo, Sep 05 2001

Keywords

• easy
• nonn

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.

Crossrefs

Cf. A034448, A063948.

Programs

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 *)

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

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.

Extensions

Edited by N. J. A. Sloane, May 04 2007