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A005237
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Numbers k such that k and k+1 have the same number of divisors.
(Formerly M2068)
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82
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2, 14, 21, 26, 33, 34, 38, 44, 57, 75, 85, 86, 93, 94, 98, 104, 116, 118, 122, 133, 135, 141, 142, 145, 147, 158, 171, 177, 189, 201, 202, 205, 213, 214, 217, 218, 230, 231, 242, 243, 244, 253, 285, 296, 298, 301, 302, 326, 332, 334, 344, 374, 375, 381, 387
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OFFSET
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1,1
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COMMENTS
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Is a(n) asymptotic to c*n with 9 < c < 10? - Benoit Cloitre, Sep 07 2002
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
I conjecture the contrary: the sequence is superlinear. Perhaps a(n) ~ n log log n. - Charles R Greathouse IV, Aug 17 2011
Erdős proved that this sequence is superlinear. Is a more specific result known? - Charles R Greathouse IV, Dec 05 2012
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
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, B18.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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EXAMPLE
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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
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MATHEMATICA
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A005237Q = DivisorSigma[0, #] == DivisorSigma[0, # + 1] &; Select[Range[387], A005237Q] (* JungHwan Min, Mar 02 2017 *)
SequencePosition[DivisorSigma[0, Range[400]], {x_, x_}][[All, 1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 25 2019 *)
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PROG
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(Python)
from sympy import divisor_count as tau
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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