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%I M2068 #88 Feb 08 2024 01:35:53
%S 2,14,21,26,33,34,38,44,57,75,85,86,93,94,98,104,116,118,122,133,135,
%T 141,142,145,147,158,171,177,189,201,202,205,213,214,217,218,230,231,
%U 242,243,244,253,285,296,298,301,302,326,332,334,344,374,375,381,387
%N Numbers k such that k and k+1 have the same number of divisors.
%C Is a(n) asymptotic to c*n with 9 < c < 10? - _Benoit Cloitre_, Sep 07 2002
%C Let S = {(n, a(n)): n is a positive integer < 2*10^5}, where {a(n)} is the above sequence. The best-fit (least squares) line through S has equation y = 9.63976*x - 1453.76. S is very linear: the square of the correlation coefficient of {n} and {a(n)} is about 0.999943. - _Joseph L. Pe_, May 15 2003
%C I conjecture the contrary: the sequence is superlinear. Perhaps a(n) ~ n log log n. - _Charles R Greathouse IV_, Aug 17 2011
%C Erdős proved that this sequence is superlinear. Is a more specific result known? - _Charles R Greathouse IV_, Dec 05 2012
%C Heath-Brown proved that this sequence is infinite. Hildebrand and Erdős, Pomerance, & Sárközy show that n sqrt(log log n) << a(n) << n (log log n)^3, where << is Vinogradov notation. - _Charles R Greathouse IV_, Oct 20 2013
%D R. K. Guy, Unsolved Problems in Number Theory, B18.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Amiram Eldar, <a href="/A005237/b005237.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 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, p. 840.
%H P. Erdős, <a href="http://www.renyi.hu/~p_erdos/1936-03.pdf">On a problem of Chowla and some related problems</a>, Proc. Cambridge Philos. Soc. 32 (1936), pp. 530-540.
%H P. Erdős, C. Pomerance, and A. Sárközy, <a href="http://www.math.dartmouth.edu/~carlp/PDF/58.pdf">On locally repeated values of certain arithmetic functions, II</a>, Acta Math. Hungarica 49 (1987), pp. 251-259. [<a href="http://www.renyi.hu/~p_erdos/1987-14.pdf">alternate link</a>]
%H D. R. Heath-Brown, <a href="http://dx.doi.org/10.1112/S0025579300010743">The divisor function at consecutive integers</a>, Mathematika 31 (1984), pp. 141-149.
%H Adolf Hildebrand, <a href="http://projecteuclid.org/euclid.pjm/1102690578">The divisor function at consecutive integers</a>, Pacific J. Math. 129:2 (1987), pp. 307-319.
%e 14 is in the sequence because 14 and 15 are both in A030513. 104 is in the sequence because 104 and 105 are both in A030626. - _R. J. Mathar_, Jan 09 2022
%t A005237Q = DivisorSigma[0, #] == DivisorSigma[0, # + 1] &; Select[Range[387], A005237Q] (* _JungHwan Min_, Mar 02 2017 *)
%t SequencePosition[DivisorSigma[0,Range[400]],{x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jan 25 2019 *)
%o (PARI) is(n)=numdiv(n)==numdiv(n+1) \\ _Charles R Greathouse IV_, Aug 17 2011
%o (Python)
%o from sympy import divisor_count as tau
%o [n for n in range(1,401) if tau(n) == tau(n+1)] # _Karl V. Keller, Jr._, Jul 10 2020
%Y Cf. A000005, A005238, A006601, A049051, A006558, A019273, A039665, A073915.
%Y Equals A083795(n-1) - 1.
%K nonn
%O 1,1
%A _N. J. A. Sloane_
%E More terms from _Jud McCranie_, Oct 15 1997
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