A073915 -id:A073915 - 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

A139571 Numbers with 31 divisors.

Original entry on oeis.org

1073741824, 205891132094649, 931322574615478515625, 22539340290692258087863249, 17449402268886407318558803753801, 2619995643649944960380551432833049

Offset: 1

Omar E. Pol, May 07 2008

30th powers of primes. The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.

Start of 31st row of A073915. - R. J. Mathar, Jun 27 2009, Jun 28 2009

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, Index entries for number of divisors

Cf. A073915, A122971, A137493 (30 divs), A175742 (32 divs).

Prime[Range[11]]^30 (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

a(n)=prime(n)^30 \\ Charles R Greathouse IV, Jun 19 2016

a(n) = A000040(n)^(31-1) = A000040(n)^30.

a(n) = A122971(A000040(n)). - R. J. Mathar, Jun 27 2009

A073917 Smallest prime which leaves a remainder 1 when divided by primorial(n), i.e., when divided by first n primes.

Original entry on oeis.org

3, 7, 31, 211, 2311, 120121, 4084081, 106696591, 892371481, 71166625531, 200560490131, 29682952539241, 2129751844690471, 78496567990020181, 8608456956238879741, 97767475431570134191, 9613801750771063195351

Offset: 1

Amarnath Murthy, Aug 18 2002

nonn

Let Pr(n) = the product of first n primes. Then a(n) is the smallest prime of the form k*Pr(n) + 1. k = 1 for first five terms.

Smallest prime p such that the prime factorization of p-1 contains the first n primes. - R. J. Mathar, Jul 03 2012

T. D. Noe, Table of n, a(n) for n = 1..150

Cf. A002110 (primorials), A073915, A103783, A214089.

Cf. A076689 (values of k).

a(n)=if(n<0,0,s=1; while(prime(s)%prod(i=1,n, prime(i))>1,s++); s)

More terms from Vladeta Jovovic, Aug 20 2002

A166546 Natural numbers n such that d(n) + 1 is prime.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 80, 82, 83, 84, 85, 86, 87, 89, 90

Offset: 1

Giovanni Teofilatto, Oct 16 2009

nonn

Natural numbers n such that d(d(n)+1)= 2. - Giovanni Teofilatto, Oct 26 2009

The complement is the union of A001248, A030514, A030516, A030626, A030627, A030629, A030631, A030632, A030633 etc. - R. J. Mathar, Oct 26 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..6955

Cf. A000005.

Cf. A073915. - R. J. Mathar, Oct 26 2009

[n: n in [1..100] | IsPrime(NumberOfDivisors(n)+1)]; // Vincenzo Librandi, Jan 20 2019

Select[Range@90, PrimeQ[DivisorSigma[0, #] + 1] &] (* Vincenzo Librandi, Jan 20 2019 *)

isok(n) = isprime(numdiv(n)+1); \\ Michel Marcus, Jan 20 2019

{1} U A000040 U A030513 U A030515 U A030628 U A030630 U A030634 U A030636 U A137485 U A137491 U A137493 U ... . - R. J. Mathar, Oct 26 2009

A073916 The n-th number with n divisors.

Original entry on oeis.org

1, 3, 25, 14, 14641, 44, 24137569, 70, 1089, 405, 819628286980801, 160, 22563490300366186081, 2752, 9801, 462, 21559177407076402401757871041, 1044, 740195513856780056217081017732809, 1520, 141376, 84992

Offset: 1

Amarnath Murthy, Aug 18 2002

nonn

Last element in n-th row of A073915.

a(p) = prime(p)^(p-1) for any prime number p. - Paul Tek, Jun 23 2013

			For n = 4; the numbers with four divisors are 6, 8, 10, 14, 15, 21,... so a(4) = 14 because 14 is the 4th number with four divisors. - _Omar E. Pol_, Jun 01 2012

T. D. Noe, Table of n, a(n) for n = 1..100

Cf. A073915.

d = Table[Length[Divisors[n]], {n, 200000}]; t = {}; n = 0; ok = True; While[ok, n++; If[PrimeQ[n], AppendTo[t, Prime[n]^(n - 1)], c = Flatten[Position[d, n, 1, n]]; If[Length[c] >= n, AppendTo[t, c[[n]]], ok = False]]]; t (* T. D. Noe, Jun 23 2013 *)

More terms from Sascha Kurz, Jan 28 2003
New name from Omar E. Pol, Jun 01 2012
a(22)-a(34) from Paul Tek, Jun 23 2013

A119586 Triangle where T(n,m) = (n+1-m)-th positive integer with (m+1) divisors.

Original entry on oeis.org

2, 3, 4, 5, 9, 6, 7, 25, 8, 16, 11, 49, 10, 81, 12, 13, 121, 14, 625, 18, 64, 17, 169, 15, 2401, 20, 729, 24, 19, 289, 21, 14641, 28, 15625, 30, 36, 23, 361, 22, 28561, 32, 117649, 40, 100, 48, 29, 529, 26, 83521, 44, 1771561, 42, 196, 80, 1024, 31, 841, 27

Offset: 1

Leroy Quet, May 31 2006

From Peter Munn, May 17 2023: (Start)
As a square array A(n,m), n, m >= 1, read by ascending antidiagonals, A(n,m) is the n-th positive integer with m+1 divisors.
Thus both formats list the numbers with m+1 divisors in their m-th column. For the corresponding sequences giving numbers with a specific number of divisors see the index entries link.
(End)

			Looking at the 4th row, 7 is the 4th positive integer with 2 divisors, 25 is the 3rd positive integer with 3 divisors, 8 is the 2nd positive integer with 4 divisors and 16 is the first positive integer with 5 divisors. So the 4th row is (7,25,8,16).
The triangle T(n,m) begins:
  n\m:    1     2     3     4     5     6     7
  ---------------------------------------------
   1 :    2
   2 :    3     4
   3 :    5     9     6
   4 :    7    25     8    16
   5 :   11    49    10    81    12
   6 :   13   121    14   625    18    64
   7 :   17   169    15  2401    20   729    24
  ...
Square array A(n,m) begins:
  n\m:     1      2      3       4      5  ...
  --------------------------------------------
   1 :     2      4      6      16     12  ...
   2 :     3      9      8      81     18  ...
   3 :     5     25     10     625     20  ...
   4 :     7     49     14    2401     28  ...
   5 :    11    121     15   14641     32  ...
  ...

Index entries for sequences related to divisors of numbers

Columns: A000040, A001248, A007422, A030514, A030515, A030516, A030626, A030627, A030628, ... (see the index entries link for more).
Cf. A073915.
Diagonals (equivalently, rows of the square array) start: A005179\{1}, A161574.
Cf. A091538.

t[n_, m_] := Block[{c = 0, k = 1}, While[c < n + 1 - m, k++; If[DivisorSigma[0, k] == m + 1, c++ ]]; k]; Table[ t[n, m], {n, 11}, {m, n}] // Flatten (* Robert G. Wilson v, Jun 07 2006 *)

More terms from Robert G. Wilson v, Jun 07 2006

A161574 a(n) = the second smallest positive integer with exactly n divisors.

Original entry on oeis.org

3, 9, 8, 81, 18, 729, 30, 100, 80, 59049, 72, 531441, 320, 324, 168, 43046721, 252, 387420489, 336, 1600, 5120, 31381059609, 420, 10000, 20480, 1764, 1344, 22876792454961, 1008, 205891132094649, 1080, 25600, 327680, 11664, 1440

Offset: 2

Leroy Quet, Jun 14 2009

nonn

Second column of A073915. [From R. J. Mathar, Jun 16 2009]

A005179

Extended by R. J. Mathar, Jun 16 2009

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

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A139571 Numbers with 31 divisors.

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A073917 Smallest prime which leaves a remainder 1 when divided by primorial(n), i.e., when divided by first n primes.

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A166546 Natural numbers n such that d(n) + 1 is prime.

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A073916 The n-th number with n divisors.

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A119586 Triangle where T(n,m) = (n+1-m)-th positive integer with (m+1) divisors.

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A161574 a(n) = the second smallest positive integer with exactly n divisors.

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