A083795 Numbers k such that k and k-1 have the same number of divisors. Numbers not included in A083794.

3, 15, 22, 27, 34, 35, 39, 45, 58, 76, 86, 87, 94, 95, 99, 105, 117, 119, 123, 134, 136, 142, 143, 146, 148, 159, 172, 178, 190, 202, 203, 206, 214, 215, 218, 219, 231, 232, 243, 244, 245, 254, 286, 297, 299, 302, 303, 327, 333, 335, 345, 375, 376, 382, 388

Offset: 1

Author

Amarnath Murthy, May 07 2003

Keywords

• nonn

Comments

Also numbers n such that sigma_0(n+1) * sigma_0(n) / (sigma_0(n+1) + sigma_0(n)) = c, c an integer. - Ctibor O. Zizka, Nov 01 2008

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

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• 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.

Crossrefs

Cf. A083794.

Programs

Maple

with(numtheory): for n from 3 to 10^3 do if tau(n) = tau(n-1) then printf(`%d,`,n) fi: od:

Mathematica

SequencePosition[DivisorSigma[0,Range[400]],{x_,x_}][[All,2]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 09 2019 *)

PARI

is(n)=n>2 && numdiv(n)==numdiv(n-1) \\ Charles R Greathouse IV, Jul 21 2015

Extensions

Corrected and extended by James Sellers, May 19 2003