%I M4950 #115 Feb 07 2024 01:14:02

%S 14,206,957,1334,1364,1634,2685,2974,4364,14841,18873,19358,20145,

%T 24957,33998,36566,42818,56564,64665,74918,79826,79833,84134,92685,

%U 109214,111506,116937,122073,138237,147454,161001,162602,166934

%N Numbers k such that k and k+1 have same sum of divisors.

%C 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

%C Whether this sequence is infinite is an unsolved problem, as noted in many of the references and links. - _Franklin T. Adams-Watters_, Jan 25 2010

%C 144806446575 is the first term for which sigma(n)/n > 2.25. All n < 10^12 have sigma(n)/n > 3/2. - _T. D. Noe_, Feb 18 2010

%C A053222(a(n)) = 0. - _Reinhard Zumkeller_, Dec 28 2011

%C Numbers n such that n + 1 = antisigma(n+1) - antisigma(n), where antisigma(n) = A024816(n) = the sum of the non-divisors of n that are between 1 and n. Example for n = 14: 15 = antisigma(15) - antisigma(14) = 96 - 81. - _Jaroslav Krizek_, Nov 10 2013

%C Up to 10^13, the value of the sigma(n)/n varies between 1417728000/945151999 (attained for n = 2835455997) and 2913242112/1263730145 (for n = 5174974943775). - _Giovanni Resta_, Feb 26 2014

%C Also numbers n such that A242962(n) = A242962(n+1), with A242962(n) = T(n) mod antisigma(n), where T(n) = A000217(n) is the n-th triangular number and antisigma(n) = A024816(n) is the sum of numbers less than n which do not divide n. - _Jaroslav Krizek_, May 29 2014

%C Guy and Shanks construct 5559060136088313 as a term of this sequence. - _Michel Marcus_, Dec 29 2014

%C Note that in all cases, n and n+1 are composite. - _Zak Seidov_, May 03 2016

%D 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.

%D R. K. Guy, Unsolved Problems in Theory of Numbers, Sect. B13.

%D W. Sierpiński, A Selection of Problems in the Theory of Numbers. Macmillan, NY, 1964, p. 110.

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

%H Giovanni Resta, <a href="/A002961/b002961.txt">Table of n, a(n) for n = 1..10135</a> (terms < 10^13; first 4804 terms from T. D. Noe)

%H 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].

%H Jonathan Bayless and Paul Kinlaw, <a href="https://www.researchgate.net/publication/288826545_On_repeated_values_of_s_and_multiperfect_numbers">On repeated values of sigma and multiperfect numbers</a>, Journal of Combinatorics and Number Theory, Vol. 7, No. 3 (2015), pp. 177-189.

%H Lourdes Benito, <a href="http://arxiv.org/abs/0707.2190">Solutions of the problem of Erdős-Sierpiński: sigma(n)=sigma(n+1)</a>, arXiv:0707.2190 [math.NT], 2007.

%H Richard Guy and Daniel Shanks, <a href="http://www.fq.math.ca/Scanned/12-3/guy.pdf">A Constructed Solution of sigma(n) = sigma(n+1)</a>, The Fibonacci Quarterly, Volume 12, Number 3, October 1974, 299.

%H A. Makowski, <a href="http://www.jstor.org/stable/2310107">On Some Equations Involving Functions phi(n) and sigma(n)</a>, The American Mathematical Monthly, Vol. 67, No. 7 (Aug. - Sep., 1960), pp. 668-670.

%H N. J. A. Sloane & D. Singmaster, <a href="/A002961/a002961.pdf">Correspondence 1972</a>.

%H Andreas Weingartner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Weingartner/wein3.html">On the Solutions of sigma(n) = sigma(n+k)</a>, Journal of Integer Sequences, Vol. 14 (2011), #11.5.5.

%F Sum_{n>=1} 1/a(n) is in the interval (0.080958, 610837) (Bayless and Kinlaw, 2015). - _Amiram Eldar_, Oct 15 2020

%t Flatten[Position[Partition[DivisorSigma[1,Range[170000]],2,1],{x_,x_}]] (* _Harvey P. Dale_, Aug 08 2011 *)

%t SequencePosition[DivisorSigma[1,Range[200000]],{x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Mar 06 2018 *)

%o (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

%o (Haskell)

%o import Data.List (elemIndices)

%o a002961 n = a002961_list !! (n-1)

%o a002961_list = map (+ 1) $ elemIndices 0 a053222_list

%o -- _Reinhard Zumkeller_, Dec 28 2011

%Y Cf. A000203 (sigma function), A053215, A053249, A054004

%Y Cf. A007373, A015861, A015863, A015865, A015866, A015867, A015876, A015877, A015880, A015881, A015882, A015883, A181647. - _Reinhard Zumkeller_, Nov 03 2010

%Y Cf. A238380.

%K nonn,nice

%O 1,1

%A _N. J. A. Sloane_, _Mira Bernstein_, _Robert G. Wilson v_

%E More terms from _Jud McCranie_, Oct 15 1997