A083795 -id:A083795 - cp's OEIS frontend

A005237 Numbers k such that k and k+1 have the same number of divisors.

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

Offset: 1

N. J. A. Sloane

nonn

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

			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

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).

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.

Cf. A000005, A005238, A006601, A049051, A006558, A019273, A039665, A073915.

Equals A083795(n-1) - 1.

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 *)

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

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

More terms from Jud McCranie, Oct 15 1997

A083794 Numbers k such that tau(k) is different from tau(k-1), where tau(m) = number of divisors of m.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 36, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82

Offset: 1

Amarnath Murthy, May 07 2003

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000
P. Erdős, On a problem of Chowla and some related problems, Proc. Cambridge Philos. Soc. 32 (1936), pp. 530-540.

Cf. A083795.

with(numtheory): for n from 1 to 150 do if tau(n) <> tau(n-1) then printf(`%d,`,n) fi: od: # James Sellers, May 19 2003

a083794[n_] :=
 Prepend[Select[Range[1, n],
DivisorSigma[0, #] != DivisorSigma[0, # - 1] &], 1]; a083794[82] (* Michael De Vlieger, Dec 24 2014 *)

PARI
```
is(n)=numdiv(n-1)!=numdiv(n)
```

Erdős proved that a(n) ~ n. - Charles R Greathouse IV, Dec 05 2012

More terms from James Sellers, May 19 2003

A146747 Numbers k such that sigma_1(k)*sigma_0(k)/(sigma_1(k)-sigma_0(k)) is an integer.

Original entry on oeis.org

2, 3, 5, 6, 7, 20, 30

Offset: 1

Ctibor O. Zizka, Nov 01 2008

Numbers k such that A000203(k)*A000005(k)/(A000203(k)-A000005(k)) is an integer.

Cf. A000005, A000203, A063647, A083795.

q[n_] := Module[{s = DivisorSigma[1, n], d = DivisorSigma[0, n]}, Divisible[s*d, s - d]]; Select[Range[2, 100], q] (* Amiram Eldar, Apr 07 2024 *)

is(n) = {my(f = factor(n), s = sigma(f), d = numdiv(f)); n > 1 && !((s*d) % (s-d));} \\ Amiram Eldar, Apr 07 2024

A146746 Duplicate of A073541.

Original entry on oeis.org

5, 14, 54, 55, 56, 58, 70, 87, 88, 89, 114, 118, 123, 125, 178, 201, 204, 206, 220, 246, 354, 358, 417, 419, 459, 478, 500, 534, 549, 594, 604, 620, 646, 654, 663, 715, 718, 753, 834, 835, 838, 894, 957

Offset: 1

Ctibor O. Zizka, Nov 01 2008

dead

A000203(n+1)*A000203(n)/(A000203(n+1)+A000203(n)) = c, c an integer.

Same as A073541. - Georg Fischer, Nov 02 2018

Cf. A083795, A063647, A073541, A000203.

sQ[n_]:=Module[{sn=DivisorSigma[1,n],sn1=DivisorSigma[1,n+1]},Divisible[sn sn1,sn+sn1]]; Select[Range[1000],sQ]  (* Harvey P. Dale, Apr 21 2011 *)

Corrected by Harvey P. Dale, Apr 21 2011

cp's OEIS Frontend

A005237 Numbers k such that k and k+1 have the same number of divisors.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A083794 Numbers k such that tau(k) is different from tau(k-1), where tau(m) = number of divisors of m.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A146747 Numbers k such that sigma_1(k)*sigma_0(k)/(sigma_1(k)-sigma_0(k)) is an integer.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

A146746 Duplicate of A073541.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions