

A002961


Numbers n such that n and n+1 have same sum of divisors.
(Formerly M4950)


51



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
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OFFSET

1,1


COMMENTS

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.  Franklin T. AdamsWatters, 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.  T. D. Noe, Feb 18 2010
A053222(a(n)) = 0.  Reinhard Zumkeller, Dec 28 2011
Numbers n such that n + 1 = antisigma(n+1)  antisigma(n), where antisigma(n) = A024816(n) = the sum of the nondivisors of n that are between 1 and n. Example for n = 14: 15 = antisigma(15)  antisigma(14) = 96  81.  Jaroslav Krizek, Nov 10 2013
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
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 nth 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
Guy and Shanks construct 5559060136088313 as a term of this sequence.  Michel Marcus, Dec 29 2014
Note that in all cases, n and n+1 are composite.  Zak Seidov, May 03 2016


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

T. D. Noe and 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 [alternative scanned copy].
Lourdes Benito, Solutions of the problem of ErdősSierpiński: sigma(n)=sigma(n+1), arXiv:0707.2190 [math.NT], 2007.
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. 668670.
A. Weingartner, On the Solutions of sigma(n) = sigma(n+k), Journal of Integer Sequences, Vol. 14 (2011), #11.5.5.


MAPLE

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


MATHEMATICA

f[n_]:=DivisorSigma[1, n]; lst={}; Do[If[f[n]==f[n+1], AppendTo[lst, n]], {n, 9!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jul 22 2009 *)
Flatten[Position[Partition[DivisorSigma[1, Range[170000]], 2, 1], {x_, x_}]] (* Harvey P. Dale, Aug 08 2011 *)


PROG

(PARI) t1=sigma(1); for(n=2, 1e6, t2=sigma(n); if(t2==t1, print1(n1", ")); t1=t2) \\ Charles R Greathouse IV, Jul 15 2011
(Haskell)
import Data.List (elemIndices)
a002961 n = a002961_list !! (n1)
a002961_list = map (+ 1) $ elemIndices 0 a053222_list
 Reinhard Zumkeller, Dec 28 2011
(MAGMA) [n: n in [3..100000]  ((n*(n+1)div 2) mod (n*(n+1)div 2SumOfDivisors(n))) eq (((n+1)*(n+2)div 2) mod ((n+1)*(n+2)div 2SumOfDivisors(n+1)))] // Jaroslav Krizek, May 29 2014


CROSSREFS

Cf. A000203 (sigma function), A053215, A053249, A054004
Cf. A007373, A015861, A015863, A015865, A015866, A015867, A015876, A015877, A015880, A015881, A015882, A015883, A181647.  Reinhard Zumkeller, Nov 03 2010
Cf. A238380.
Sequence in context: A068769 A113349 A109764 * A063071 A251963 A192007
Adjacent sequences: A002958 A002959 A002960 * A002962 A002963 A002964


KEYWORD

nonn,nice


AUTHOR

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v


EXTENSIONS

More terms from Jud McCranie, Oct 15 1997


STATUS

approved



