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A002961
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Numbers n such that n and n+1 have same sum of divisors.
(Formerly M4950)
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43
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14, 206, 957, 1334, 1364, 1634, 2685, 2974, 4364, 14841, 18873, 19358, 20145, 24957, 33998, 36566, 42818, 56564, 64665, 74918, 79826, 79833, 84134, 92685, 109214, 111506, 116937, 122073, 138237, 147454, 161001, 162602, 166934
(list;
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refs;
listen;
history;
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OFFSET
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1,1
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COMMENTS
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For the values of n < 2*10^10 in this sequence, sigma(n)/n is between 1.5 and 2.25. - T. D. Noe, Sep 17 2007
Whether this sequence is infinite is an unsolved problem, as noted in many of the references and links. [From Franklin T. Adams-Watters, Jan 25 2010]
144806446575 is the first term for which sigma(n)/n > 2.25. All n < 10^12 have sigma(n)/n > 3/2. [From T. D. Noe, Feb 18 2010]
A053222(a(n)) = 0. [Reinhard Zumkeller, Dec 28 2011]
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
R. K. Guy, Unsolved Problems in Theory of Numbers, Sect. B13.
W. Sierpi\'{n}ski, A Selection of Problems in the Theory of Numbers. Macmillan, NY, 1964, p. 110.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. Weingartner, On the Solutions of sigma(n) = sigma(n+k), Journal of Integer Sequences, Vol. 14 (2011), #11.5.5.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..4804 (terms < 10^12)
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].
Lourdes Benito, Solutions of the problem of Erdos-Sierpinski: sigma(n)=sigma(n+1)
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MAPLE
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with(numtheory); P:=proc(n) local a, i; for i from 1 by 1 to n do a:=sigma(i)/sigma(i+1); if a=1 then print(i); fi; od; end: P(100000); - Paolo P. Lava, Aug 23 2007
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MATHEMATICA
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f[n_]:=DivisorSigma[1, n]; lst={}; Do[If[f[n]==f[n+1], AppendTo[lst, n]], {n, 9!}]; lst [From Vladimir Orlovsky, Jul 22 2009]
Flatten[Position[Partition[DivisorSigma[1, Range[170000]], 2, 1], {x_, x_}]] (* From Harvey P. Dale, Aug 08 2011 *)
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PROG
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(PARI) t1=sigma(1); for(n=2, 1e6, t2=sigma(n); if(t2==t1, print1(n-1", ")); t1=t2) \\ Charles R Greathouse IV, Jul 15 2011
(Haskell)
import Data.List (elemIndices)
a002961 n = a002961_list !! (n-1)
a002961_list = map (+ 1) $ elemIndices 0 a053222_list
-- Reinhard Zumkeller, Dec 28 2011
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CROSSREFS
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A000203 (sigma function), A053215, A053249, A054004
Cf. A007373, A015861, A015863, A015865, A015866, A015867, A015876, A015877, A015880, A015881, A015882, A015883, A181647. [From Reinhard Zumkeller, Nov 03 2010]
Sequence in context: A068769 A113349 A109764 * A063071 A192007 A160682
Adjacent sequences: A002958 A002959 A002960 * A002962 A002963 A002964
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v
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EXTENSIONS
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More terms from Jud McCranie Oct 15 1997
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STATUS
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approved
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