

A005237


Numbers n such that n and n+1 have the same number of divisors.
(Formerly M2068)


66



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

1,1


COMMENTS

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 bestfit (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
HeathBrown 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


REFERENCES

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).


LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 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, p. 840.
P. Erdős, On a problem of Chowla and some related problems, Proc. Cambridge Philos. Soc. 32 (1936), pp. 530540.
P. Erdős, C. Pomerance, and A. Sárközy, On locally repeated values of certain arithmetic functions, II, Acta Math. Hungarica 49 (1987), pp. 251259. [alternate link]
D. R. HeathBrown, The divisor function at consecutive integers, Mathematika 31 (1984), pp. 141149.
Adolf Hildebrand, The divisor function at consecutive integers, Pacific J. Math. 129:2 (1987), pp. 307319.


MAPLE

with(numtheory); A005237:=proc(q) local n;
for n from 1 to q do if tau(n)=tau(n+1) then print(n); fi;
od; end: A005237(10^4); # Paolo P. Lava, May 03 2013


MATHEMATICA

f[n_]:=Length[Divisors[n]]; lst={}; Do[If[f[n]==f[n+1], AppendTo[lst, n]], {n, 7!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 14 2009 *)
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 *)


PROG

(PARI) is(n)=numdiv(n)==numdiv(n+1) \\ Charles R Greathouse IV, Aug 17 2011
(Python)
from sympy import divisor_count as tau
[n for n in range(1, 401) if tau(n) == tau(n+1)] # Karl V. Keller, Jr., Jul 10 2020


CROSSREFS

Cf. A000005, A005238, A006601, A049051, A006558, A019273, A039665.
Equals A083795(n1)  1.
Sequence in context: A101398 A131221 A138047 * A140578 A052213 A280074
Adjacent sequences: A005234 A005235 A005236 * A005238 A005239 A005240


KEYWORD

nonn


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from Jud McCranie, Oct 15 1997


STATUS

approved



