OFFSET
1,1
COMMENTS
Complement of A078923. - Lekraj Beedassy, Jul 19 2005
Chen & Zhao show that the lower density of this sequence is at least 0.06, improving on te Riele. - Charles R Greathouse IV, Dec 28 2013
Numbers k such that A048138(k) = 0. A048138(k) measures how "touchable" k is. - Jeppe Stig Nielsen, Jan 12 2020
From Amiram Eldar, Feb 13 2021: (Start)
The term "untouchable number" was coined by Alanen (1972). He found the 570 terms below 5000.
Erdős (1973) proved that the lower asymptotic density of untouchable numbers is positive, te Riele (1976) proved that it is > 0.0324, and Banks and Luca (2004, 2005) proved that it is > 1/48.
Pollack and Pomerance (2016) conjectured that the asymptotic density is ~ 0.17. (End)
The upper asymptotic density is less than 1/2 by the 'almost all' binary Goldbach conjecture, independently proved by Nikolai Chudakov, Johannes van der Corput, and Theodor Estermann. (In this context, this shows that the density of the odd numbers of this form is 0 (consider A001065(p*q) for prime p, q); full Goldbach would prove that 5 is the only odd number in this sequence.) - Charles R Greathouse IV, Dec 05 2022
REFERENCES
Richard K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004, section B10, pp. 100-101.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 65.
József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 93.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 125.
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..13863 (terms < 10^5; first 8153 terms from Klaus Brockhaus)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], p. 840.
Jack David Alanen, Empirical study of aliquot series, Ph.D Thesis, Yale University, 1972.
William D. Banks and Florian Luca, Noncototients and Nonaliquots, arXiv:math/0409231 [math.NT], 2004.
William D. Banks and Florian Luca, Nonaliquots and Robbins numbers, Colloq. Math., Vol. 103, No. 1 (2005), pp. 27-32.
Yong-Gao Chen and Qing-Qing Zhao, Nonaliquot numbers, Publ. Math. Debrecen, Vol. 78, No. 2 (2011), pp. 439-442.
K. Chum, R. K. Guy, M. J. Jacobson, Jr., and A. S. Mosunov, Numerical and statistical analysis of aliquot sequences, Experimental Mathematics (2018), pp. 1-12.
Paul Erdős, Über die Zahlen der Form sigma(n)-n und n-phi(n), Elemente der Math., Vol. 28 (1973), pp. 83-86; alternative link (in German).
Victor Meally, Letter to N. J. A. Sloane, no date.
Paul Pollack, Not Always Buried Deep: A Second Course in Elementary Number Theory, AMS, 2009, p. 272.
Paul Pollack and Carl Pomerance, Some problems of Erdős on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B, Vol. 3 (2016), pp. 1-26; Errata.
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, Math. Comp., Vol. 83, No. 288 (2014), pp. 1903-1913; alternative link.
Giovanni Resta, Untouchable numbers (the 150232 terms up to 10^6).
H. J. J. te Riele, A theoretical and computational study of generalized aliquot sequences, Mathematisch Centrum, Amsterdam, 1976. See chapter 9.
Eric Weisstein's World of Mathematics, Untouchable Number.
Wikipedia, Untouchable number.
Robert G. Wilson v, Letter to N. J. A. Sloane, Jul. 1992.
MATHEMATICA
untouchableQ[n_] := Catch[ Do[ If[n == DivisorSigma[1, k]-k, Throw[True]], {k, 0, (n-1)^2}]] === Null; Reap[ Table[ If[ untouchableQ[n], Print[n]; Sow[n]], {n, 2, 700}]][[2, 1]] (* Jean-François Alcover, Jun 29 2012, after Benoit Cloitre *)
PROG
(PARI) isA078923(n)=if(n==0 || n==1, return(1)); for(m=1, (n-1)^2, if( sigma(m)-m == n, return(1))); 0
isA005114(n)=!isA078923(n)
for(n=1, 700, if (isA005114(n), print(n))) \\ R. J. Mathar, Aug 10 2006
(PARI) is(n)=if(n%2 && n<4e18, return(n==5)); forfactored(m=1, (n-1)^2, if(sigma(m)-m[1]==n, return(0))); 1 \\ Charles R Greathouse IV, Dec 05 2022
(Python)
from sympy import divisor_sigma as sigma
from functools import cache
@cache
def f(m): return sigma(m)-m
def okA005114(n):
if n < 2: return 0
return not any(f(m) == n for m in range(1, (n-1)**2+1))
print([k for k in range(289) if okA005114(k)]) # Michael S. Branicky, Nov 16 2024
(Python) # faster for intial segment of sequence
from itertools import count, islice
from sympy import divisor_sigma as sigma
def agen(): # generator of terms
n, touchable, t = 2, {0, 1}, 1
for m in count(2):
touchable.add(sigma(m)-m)
while m > t:
if n not in touchable:
yield n
else:
touchable.discard(n)
n += 1
t = (n-1)**2
print(list(islice(agen(), 20))) # Michael S. Branicky, Nov 16 2024
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from David W. Wilson
STATUS
approved