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A005237 Numbers n such that n and n+1 have the same number of divisors.
(Formerly M2068)
58
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 (list; graph; refs; listen; history; text; internal format)
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 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

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

T. D. Noe, Table of n, a(n) for n = 1..1000

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. 530-540.

P. Erdős, C. Pomerance, and A. Sárközy, On locally repeated values of certain arithmetic functions, II, Acta Math. Hungarica 49 (1987), pp. 251-259. [alternate link]

D. R. Heath-Brown, The divisor function at consecutive integers, Mathematika 31 (1984), pp. 141-149.

Adolf Hildebrand, The divisor function at consecutive integers, Pacific J. Math. 129:2 (1987), pp. 307-319.

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 *)

PROG

(PARI) is(n)=numdiv(n)==numdiv(n+1) \\ Charles R Greathouse IV, Aug 17 2011

CROSSREFS

Cf. A000005, A005238, A006601, A049051, A006558, A019273, A039665.

Equals A083795(n-1) - 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

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Last modified March 26 21:00 EDT 2017. Contains 284137 sequences.