A005114 Untouchable numbers, also called nonaliquot numbers: impossible values for the sum of aliquot parts function (A001065).

2, 5, 52, 88, 96, 120, 124, 146, 162, 188, 206, 210, 216, 238, 246, 248, 262, 268, 276, 288, 290, 292, 304, 306, 322, 324, 326, 336, 342, 372, 406, 408, 426, 430, 448, 472, 474, 498, 516, 518, 520, 530, 540, 552, 556, 562, 576, 584, 612, 624, 626, 628, 658

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

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).
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.
• 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).
• Shyam Sunder Gupta, Beauty of Number 153, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 15, 399-410.
• 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.

Crossrefs

Cf. A001065, A048138, A057709, A064000, A078923, A152454, A231964, A283152, A284147.

Programs

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 *)

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

Extensions

More terms from David W. Wilson