OFFSET
1,1
COMMENTS
For the values of k < 2*10^10 in this sequence, sigma(k)/k 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. - Franklin T. Adams-Watters, Jan 25 2010
144806446575 is the first term for which sigma(k)/k > 2.25. All k < 10^12 have sigma(k)/k > 3/2. - T. D. Noe, Feb 18 2010 [Bayless and Kinlaw (2015) proved that sigma(k)/k > 3/2 for all terms k in this sequence. - Amiram Eldar, Aug 06 2026]
Numbers k such that k + 1 = antisigma(k+1) - antisigma(k), where antisigma(k) = A024816(k) = the sum of the non-divisors of k that are between 1 and k. Example for k = 14: 15 = antisigma(15) - antisigma(14) = 96 - 81. - Jaroslav Krizek, Nov 10 2013
Up to 10^13, the value of the sigma(k)/k varies between 1417728000/945151999 (attained for k = 2835455997) and 2913242112/1263730145 (for k = 5174974943775). - Giovanni Resta, Feb 26 2014
Also numbers k such that A242962(k) = A242962(k+1), with A242962(k) = T(k) mod antisigma(k), where T(k) = A000217(k) is the k-th triangular number and antisigma(k) = A024816(k) is the sum of numbers less than k which do not divide k. - Jaroslav Krizek, May 29 2014
Guy and Shanks construct 5559060136088313 as a term of this sequence. - Michel Marcus, Dec 29 2014
Note that in all cases, k and k+1 are composite. - Zak Seidov, May 03 2016
The number of terms not exceeding x for large enough x is at most x/exp(log(x)^(1/3)) (Erdős et al., 1987). Therefore, the asymptotic density of this sequence is 0. - Amiram Eldar, Aug 06 2026
REFERENCES
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.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B13, pp. 103-104.
Wacław Sierpiński, 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).
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..10135 (terms < 10^13; first 4804 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
Jonathan Bayless and Paul Kinlaw, On repeated values of sigma and multiperfect numbers, Journal of Combinatorics and Number Theory, Vol. 7, No. 3 (2015), pp. 177-189.
Lourdes Benito, Solutions of the problem of Erdős-Sierpiński: sigma(n)=sigma(n+1), arXiv:0707.2190 [math.NT], 2007.
Paul Erdős, Carl Pomerance, and András Sárközy, On locally repeated values of certain arithmetic functions, II, Acta Mathematica Hungarica, Vol. 49, No. 1-2 (1987), pp. 251-259; alternative link.
Richard Guy and Daniel Shanks, A Constructed Solution of sigma(n) = sigma(n+1), The Fibonacci Quarterly, Volume 12, Number 3, October 1974, 299.
A. Makowski, On Some Equations Involving Functions phi(n) and sigma(n), The American Mathematical Monthly, Vol. 67, No. 7 (Aug. - Sep., 1960), pp. 668-670.
N. J. A. Sloane & D. Singmaster, Correspondence 1972.
Andreas Weingartner, On the Solutions of sigma(n) = sigma(n+k), Journal of Integer Sequences, Vol. 14 (2011), #11.5.5.
FORMULA
A053222(a(n)) = 0. - Reinhard Zumkeller, Dec 28 2011
Sum_{n>=1} 1/a(n) is in the interval (0.080958, 610837) (Bayless and Kinlaw, 2015). - Amiram Eldar, Oct 15 2020
MATHEMATICA
Flatten[Position[Partition[DivisorSigma[1, Range[170000]], 2, 1], {x_, x_}]] (* Harvey P. Dale, Aug 08 2011 *)
(* Alternative: *)
SequencePosition[DivisorSigma[1, Range[200000]], {x_, x_}][[All, 1]] (* Harvey P. Dale, Mar 06 2018 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,nice
EXTENSIONS
More terms from Jud McCranie, Oct 15 1997
STATUS
approved