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

%I A005114 M1552 #126 Aug 11 2025 08:32:03
%S A005114 2,5,52,88,96,120,124,146,162,188,206,210,216,238,246,248,262,268,276,
%T A005114 288,290,292,304,306,322,324,326,336,342,372,406,408,426,430,448,472,
%U A005114 474,498,516,518,520,530,540,552,556,562,576,584,612,624,626,628,658
%N A005114 Untouchable numbers, also called nonaliquot numbers: impossible values for the sum of aliquot parts function (A001065).
%C A005114 Complement of A078923. - _Lekraj Beedassy_, Jul 19 2005
%C A005114 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
%C A005114 Numbers k such that A048138(k) = 0. A048138(k) measures how "touchable" k is. - _Jeppe Stig Nielsen_, Jan 12 2020
%C A005114 From _Amiram Eldar_, Feb 13 2021: (Start)
%C A005114 The term "untouchable number" was coined by Alanen (1972). He found the 570 terms below 5000.
%C A005114 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.
%C A005114 Pollack and Pomerance (2016) conjectured that the asymptotic density is ~ 0.17. (End)
%C A005114 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
%D A005114 Richard K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004, section B10, pp. 100-101.
%D A005114 Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 65.
%D A005114 József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 93.
%D A005114 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A005114 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.
%D A005114 David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 125.
%H A005114 Giovanni Resta, <a href="/A005114/b005114.txt">Table of n, a(n) for n = 1..13863</a> (terms < 10^5; first 8153 terms from Klaus Brockhaus)
%H A005114 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], p. 840.
%H A005114 Jack David Alanen, <a href="https://ir.cwi.nl/pub/9143">Empirical study of aliquot series</a>, Ph.D Thesis, Yale University, 1972.
%H A005114 William D. Banks and Florian Luca, <a href="https://arxiv.org/abs/math/0409231">Noncototients and Nonaliquots</a>, arXiv:math/0409231 [math.NT], 2004.
%H A005114 William D. Banks and Florian Luca, <a href="https://www.impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/colloquium-mathematicum/all/103/1/87562/nonaliquots-and-robbins-numbers">Nonaliquots and Robbins numbers</a>, Colloq. Math., Vol. 103, No. 1 (2005), pp. 27-32.
%H A005114 Yong-Gao Chen and Qing-Qing Zhao, <a href="https://doi.org/10.5486/pmd.2011.4820">Nonaliquot numbers</a>, Publ. Math. Debrecen, Vol. 78, No. 2 (2011), pp. 439-442.
%H A005114 K. Chum, R. K. Guy, M. J. Jacobson, Jr., and A. S. Mosunov, <a href="https://doi.org/10.1080/10586458.2018.1477077">Numerical and statistical analysis of aliquot sequences</a>, Experimental Mathematics (2018), pp. 1-12.
%H A005114 Paul Erdős, <a href="http://www.renyi.hu/~p_erdos/1973-27.pdf">Über die Zahlen der Form sigma(n)-n und n-phi(n)</a>, Elemente der Math., Vol. 28 (1973), pp. 83-86; <a href="https://gdz.sub.uni-goettingen.de/id/PPN378850199_0028">alternative link</a> (in German).
%H A005114 Shyam Sunder Gupta, <a href="https://doi.org/10.1007/978-981-97-2465-9_15">Beauty of Number 153</a>, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 15, 399-410.
%H A005114 Victor Meally, <a href="/A006556/a006556.pdf">Letter to N. J. A. Sloane</a>, no date.
%H A005114 Paul Pollack, <a href="http://pollack.uga.edu/NABDofficial.pdf">Not Always Buried Deep: A Second Course in Elementary Number Theory</a>, AMS, 2009, p. 272.
%H A005114 Paul Pollack and Carl Pomerance, <a href="https://doi.org/10.1090/btran/10">Some problems of Erdős on the sum-of-divisors function</a>, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B, Vol. 3 (2016), pp. 1-26; <a href="http://pollack.uga.edu/reversal-errata.pdf">Errata</a>.
%H A005114 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 A005114 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>, Math. Comp., Vol. 83, No. 288 (2014), pp. 1903-1913; <a href="http://www.math.dartmouth.edu/~carlp/uupaper6.pdf">alternative link</a>.
%H A005114 Giovanni Resta, <a href="http://www.numbersaplenty.com/set/untouchable_number/">Untouchable numbers</a> (the 150232 terms up to 10^6).
%H A005114 H. J. J. te Riele, <a href="https://ir.cwi.nl/pub/13093">A theoretical and computational study of generalized aliquot sequences</a>, Mathematisch Centrum, Amsterdam, 1976. See chapter 9.
%H A005114 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UntouchableNumber.html">Untouchable Number</a>.
%H A005114 Wikipedia, <a href="http://en.wikipedia.org/wiki/Untouchable_number">Untouchable number</a>.
%H A005114 Robert G. Wilson v, <a href="/A007015/a007015.pdf">Letter to N. J. A. Sloane, Jul. 1992</a>.
%t A005114 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_ *)
%o A005114 (PARI) isA078923(n)=if(n==0 || n==1, return(1)); for(m=1,(n-1)^2, if( sigma(m)-m == n, return(1))); 0
%o A005114 isA005114(n)=!isA078923(n)
%o A005114 for(n=1,700, if (isA005114(n), print(n))) \\ _R. J. Mathar_, Aug 10 2006
%o A005114 (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
%o A005114 (Python)
%o A005114 from sympy import divisor_sigma as sigma
%o A005114 from functools import cache
%o A005114 @cache
%o A005114 def f(m): return sigma(m)-m
%o A005114 def okA005114(n):
%o A005114     if n < 2: return 0
%o A005114     return not any(f(m) == n for m in range(1, (n-1)**2+1))
%o A005114 print([k for k in range(289) if okA005114(k)]) # _Michael S. Branicky_, Nov 16 2024
%o A005114 (Python) # faster for intial segment of sequence
%o A005114 from itertools import count, islice
%o A005114 from sympy import divisor_sigma as sigma
%o A005114 def agen(): # generator of terms
%o A005114     n, touchable, t = 2, {0, 1}, 1
%o A005114     for m in count(2):
%o A005114         touchable.add(sigma(m)-m)
%o A005114         while m > t:
%o A005114             if n not in touchable:
%o A005114                 yield n
%o A005114             else:
%o A005114                 touchable.discard(n)
%o A005114             n += 1
%o A005114             t = (n-1)**2
%o A005114 print(list(islice(agen(), 20))) # _Michael S. Branicky_, Nov 16 2024
%Y A005114 Cf. A001065, A048138, A057709, A064000, A078923, A152454, A231964, A283152, A284147.
%K A005114 nonn,nice
%O A005114 1,1
%A A005114 _N. J. A. Sloane_
%E A005114 More terms from _David W. Wilson_