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A005114
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Untouchable numbers, also called nonaliquot numbers: impossible values for the sum of aliquot parts function (A001065).
(Formerly M1552)
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47
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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;
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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Yong-Gao Chen and Qing-Qing Zhao, Nonaliquot numbers, Publ. Math. Debrecen 78:2 (2011), pp. 439-442 (DOI: 10.5486/pmd.2011.4820).
R. K. Guy, Unsolved Problems in Number Theory, B10.
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).
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LINKS
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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 (scanned copy 11MB).
P. Erdős, Über die Zahlen der Form sigma(n)-n und n-phi(n), Elemente der Math. 28 (1973), pp. 83-86.
Victor Meally, Letter to N. J. A. Sloane, no date.
P. Pollack, C. Pomerance, Some problems of Erdos on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, 2015, to appear.
C. Pomerance and H.-S. Yang, On untouchable numbers and related problems, 2012.
C. Pomerance and H.-S. Yang, Variant of a theorem of Erdos on the sum-of-proper-divisors function, 2012.
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 (Dissertation), Mathematisch Centrum, Amsterdam, 1975 (scanned copy 13MB).
Eric Weisstein's World of Mathematics, Untouchable Number.
Wikipedia, Untouchable number
R. G. Wilson, V, Letter to N. J. A. Sloane, Jul. 1992
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FORMULA
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Complement of A078923. - Lekraj Beedassy, Jul 19 2005
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from David W. Wilson
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STATUS
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approved
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