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A005114 Untouchable numbers, also called nonaliquot numbers: impossible values for the sum of aliquot parts function (A001065).
(Formerly M1552)
50
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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)

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.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.

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.

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 93.

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.

R. G. Wilson, V, Letter to N. J. A. Sloane, Jul. 1992.

FORMULA

Complement of A078923. - Lekraj Beedassy, Jul 19 2005

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

CROSSREFS

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

Sequence in context: A208797 A004098 A208206 * A216079 A206584 A268286

Adjacent sequences:  A005111 A005112 A005113 * A005115 A005116 A005117

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from David W. Wilson

STATUS

approved

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Last modified October 21 19:16 EDT 2021. Contains 348155 sequences. (Running on oeis4.)