A005237 - OEIS

A005237

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

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

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. 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.

EXAMPLE

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

MATHEMATICA

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, A073915.

Equals A083795(n-1) - 1.

Sequence in context: A101398 A131221 A138047 * A140578 A052213 A280074