A005238 -id:A005238 - cp's OEIS frontend

A039665 Sets of 4 consecutive numbers with equal number of divisors.

242, 243, 244, 245, 3655, 3656, 3657, 3658, 4503, 4504, 4505, 4506, 5943, 5944, 5945, 5946, 6853, 6854, 6855, 6856, 7256, 7257, 7258, 7259, 8392, 8393, 8394, 8395, 9367, 9368, 9369, 9370, 10983, 10984, 10985, 10986, 11605, 11606, 11607, 11608

Offset: 1

Felice Russo

nonn

Taking the first entry in each set gives A006601.

D. Wells, Curious and interesting numbers, Penguin Books, p. 134

Cf. A000005, A005237, A005238, A006601, A049051, A006558, A019273.

More terms from Patrick De Geest, Nov 15 1999

A169834 Numbers k such that d(k-1) = d(k) = d(k+1).

Original entry on oeis.org

34, 86, 94, 142, 202, 214, 218, 231, 243, 244, 302, 375, 394, 446, 604, 634, 664, 698, 903, 922, 1042, 1106, 1138, 1262, 1275, 1310, 1335, 1346, 1402, 1642, 1762, 1833, 1838, 1886, 1894, 1925, 1942, 1982, 2014, 2055, 2102, 2134, 2182, 2218, 2265, 2306, 2344, 2362

Offset: 1

N. J. A. Sloane, Jun 02 2010

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Reinhard Zumkeller)
Erich Friedman, What's Special About This Number? (See entry 34.)

Cf. A000005, A005238.

Cf. A051950.

a169834 n = a169834_list !! (n-1)
a169834_list = f a051950_list [0..] where
   f (0:0:ws) (x:y:zs) = y : f (0:ws) (y:zs)
   f (:v:ws) (:y:zs) = f (v:ws) (y:zs)
-- Reinhard Zumkeller, Aug 31 2014

q:= n-> is(nops(map(numtheory[tau], {$n-1..n+1}))=1):
select(q, [$1..3000])[];  # Alois P. Heinz, Jun 24 2021

d[n_] := DivisorSigma[0, n];
samedQ[n_] := d[n-1] == d[n] == d[n+1];
Select[Range[3000], samedQ] (* Jean-François Alcover, Aug 01 2018 *)
1 + Flatten@Position[Differences@#&/@Partition[DivisorSigma[0, Range@3000], 3, 1], {0, 0}] (* Hans Rudolf Widmer, Feb 02 2023 *)

from sympy import divisor_count as d
def ok(n): return d(n-1) == d(n) == d(n+1)
print(list(filter(ok, range(1, 2400)))) # Michael S. Branicky, Jun 24 2021

a(n) = A005238(n) + 1. - Jon Maiga / Georg Fischer, Jun 24 2021

A049053 Numbers k such that k through k+6 all have the same number of divisors.

Original entry on oeis.org

171893, 180965, 647381, 1039493, 1071829, 1450261, 1563653, 1713413, 2129029, 2384101, 4704581, 4773301, 5440853, 5775365, 6627061, 6644405, 6697253, 8556661, 8833429, 10531253, 12101509, 12238453, 12307141, 13416661, 13970405

Offset: 1

David W. Wilson

nonn

Allan Swett found that the first term not congruent to 5 mod 16 is 67073285. - Ralf Stephan, Nov 15 2004

Since A119479(n) < 7 for n < 8, no term has fewer than 8 divisors; the first that has more is a(30)=17476613. - Ivan Neretin, Feb 05 2016

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Donovan Johnson)
Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, page 244.

Cf. A000005, A006558, A019273, A119479.

Other runs of equidivisor numbers: A005237 (runs of 2), A005238 (runs of 3), A006601 (runs of 4), A049051 (runs of 5), A049052 (runs of 6).

isok(n) = {my(nb = numdiv(n)); for (k=1, 6, if (numdiv(n+k) != nb, return (0));); 1;} \\ Michel Marcus, Feb 06 2016

A049052 Numbers k such that k through k+5 all have the same number of divisors.

Original entry on oeis.org

28374, 90181, 157493, 171893, 171894, 180965, 180966, 210133, 298694, 346502, 369061, 376742, 610310, 647381, 647382, 707286, 729542, 769862, 1039493, 1039494, 1071829, 1071830, 1243541, 1302005, 1449605, 1450261, 1450262

Offset: 1

David W. Wilson

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Donovan Johnson)

Cf. A000005, A006558, A019273, A119479.

Other runs of equidivisor numbers: A005237 (runs of 2), A005238 (runs of 3), A006601 (runs of 4), A049051 (runs of 5), A049053 (runs of 7).

SequencePosition[DivisorSigma[0,Range[1451000]],{x_,x_,x_,x_,x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 03 2020 *)

A075032 Numbers n such that tau(n) <= tau(n+1) <= tau(n+2) where tau(n) = number of divisors of n.

Original entry on oeis.org

1, 2, 13, 14, 25, 26, 33, 34, 37, 38, 43, 61, 62, 73, 74, 85, 86, 93, 94, 97, 98, 103, 115, 118, 121, 122, 133, 134, 141, 142, 145, 146, 157, 158, 163, 187, 188, 193, 194, 201, 202, 205, 206, 213, 214, 217, 218, 229, 230, 241, 242, 243, 244, 253, 254, 274, 277

Offset: 1

Amarnath Murthy, Sep 02 2002

nonn

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000

Cf. A000005, A075033, A075034, A075035, A005238 (subsequence).

from sympy import divisor_count as tau
[n for n in range(1,303) if tau(n) <= tau(n+1) <= tau(n+2)] # Karl V. Keller, Jr., Jul 10 2020

Corrected and extended by Benoit Cloitre, Sep 07 2002

A292580 T(n,k) is the start of the first run of exactly k consecutive integers having exactly 2n divisors. Table read by rows.

Original entry on oeis.org

5, 2, 6, 14, 33, 12, 44, 603, 242, 10093613546512321, 24, 104, 230, 3655, 11605, 28374, 171893, 48, 2511, 7939375, 60, 735, 1274, 19940, 204323, 368431323, 155385466971, 18652995711772, 15724736975643, 2973879756088065948, 9887353188984012120346

Offset: 1

Jon E. Schoenfield, Sep 19 2017

The number of terms in row n is A119479(2n).
Düntsch and Eggleton (1989) has typos for T(3,5) and T(10,3) (called D(6,5) and D(20,3) in their notation). Letsko (2015) and Letsko (2017) both have a wrong value for T(7,3).
The first value required to extend the data is T(6,13) <= 586683019466361719763403545; the first unknown value that may exist is T(12,19). See the a-file for other known values and upper bounds up to T(50,7).

			T(1,1) = 5 because 5 is the start of the first "run" of exactly 1 integer having exactly 2*1=2 divisors (5 is the first prime p such that both p-1 and p+1 are nonprime);
T(1,2) = 2 because 2 is the start of the first run of exactly 2 consecutive integers having exactly 2*1=2 divisors (2 and 3 are the only consecutive integers that are prime);
T(3,4) = 242 because the first run of exactly 4 consecutive integers having exactly 2*3=6 divisors is 242 = 2*11^2, 243 = 3^5, 244 = 2^2*61, 245 = 5*7^2.
Table begins:
   n  T(n,1), T(n,2), ...
  ==  ========================================================
   1  5, 2;
   2  6, 14, 33;
   3  12, 44, 603, 242, 10093613546512321;
   4  24, 104, 230, 3655, 11605, 28374, 171893;
   5  48, 2511, 7939375;
   6  60, 735, 1274, 19940, 204323, 368431323, 155385466971, 18652995711772, 15724736975643, 2973879756088065948, 9887353188984012120346, 120402988681658048433948, T(6,13), ...;
   7  192, 29888, 76571890623;
   8  120, 2295, 8294, 153543, 178086, 5852870, 17476613;
   9  180, 6075, 959075, 66251139635486389922, T(9,5);
  10  240, 5264, 248750, 31805261872, 1428502133048749, 8384279951009420621, 189725682777797295066519373;
  11  3072, 2200933376, 104228508212890623;
  12  360, 5984, 72224, 2919123, 15537948, 973277147, 33815574876, 1043710445721, 2197379769820, 2642166652554075, 17707503256664346, T(12,12), ...;
  13  12288, 689278976, 1489106237081787109375;
  14  960, 156735, 23513890624, 4094170438109373, 55644509293039461218749, 4230767238315793911295500109374, 273404501868270838132985214432619890621;
  15  720, 180224, 145705879375, 10868740069638250502059754282498, T(15,5);
  16  840, 21735, 318680, 6800934, 57645182, 1194435205, 14492398389;
  ...

Hugo van der Sanden, Table of n, a(n) for n = 1..32
Ivo Düntsch and Roger B. Eggleton, Equidivisible consecutive integers, 1989.
Vladimir A. Letsko, Some new results on consecutive equidivisible integers, arXiv:1510.07081 [math.NT], 2015.
Vladimir A. Letsko, Table of a(n) for all even n such that exact value of a(n) is proved, 2017.
Carlos Rivera, Problem 20: k consecutive numbers with the same number of divisors, The Prime Puzzles and Problems Connection.
Hugo van der Sanden, calculation of T(6,11).
Hugo van der Sanden, calculation of T(6,12).
Hugo van der Sanden, A-file with known values and bounds up to T(50,7)

Cf. A000005, A006558, A072507, A119479.
Cf. A005237, A005238, A006601, A049051, A049052, A049053.
Cf. A003680, A075036, A075040.

T(n,2) = A075036(n). - Jon E. Schoenfield, Sep 23 2017

a(1)-a(25) from Düntsch and Eggleton (1989) with corrections by Jon E. Schoenfield, Sep 19 2017
a(26)-a(27) from Giovanni Resta, Sep 20 2017
a(28)-a(29) from Hugo van der Sanden, Jan 12 2022
a(30) from Hugo van der Sanden, Sep 03 2022
a(31) added by Hugo van der Sanden, Dec 05 2022; see "calculation of T(6,11)" link for a list of the people involved.
a(32) added by Hugo van der Sanden, Dec 18 2022; see "calculation of T(6,12)" link for a list of the people involved.

A332313 Numbers k such that k, k + 1 and k + 2 have the same number of divisors in Gaussian integers.

Original entry on oeis.org

23824, 38574, 52974, 62224, 71406, 105424, 110574, 191824, 201616, 209424, 240174, 249775, 282896, 285102, 297774, 326574, 340974, 375824, 393424, 407824, 440656, 451024, 496174, 509776, 553774, 587536, 599632, 600174, 606032, 623824, 628974, 631376, 667024, 672174

Offset: 1

Amiram Eldar, Feb 09 2020

nonn

			23824 is a term since 23824, 23825 and 23826 each have 36 divisors in Gaussian integers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005238, A062327, A332312, A332314.

Flatten[Position[Partition[DivisorSigma[0, Range[3*10^5], GaussianIntegers -> True], 3, 1], {x_, x_, x_}]] (* after Harvey P. Dale at A005238 *)

A338453 Starts of runs of 3 consecutive numbers with the same total binary weight of their divisors (A093653).

Original entry on oeis.org

3, 242, 243, 1837, 2361, 3693, 3728, 6061, 6457, 9782, 11181, 11721, 13855, 15177, 20017, 22591, 28021, 31461, 31887, 33098, 33993, 38137, 52016, 52112, 60321, 76897, 78542, 78745, 80461, 108394, 116017, 119541, 124453, 125493, 127117, 127417, 145369, 151805, 154113

Offset: 1

Amiram Eldar, Oct 28 2020

Numbers k such that A093653(k) = A093653(k+1) = A093653(k+2).

			3 is a term since A093653(3) = A093653(4) = A093653(5) = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A093653.
Subsequence of A338452.
Similar sequences: A005238, A006073, A045939.

f[n_] := DivisorSum[n, DigitCount[#, 2, 1] &]; s = {}; m = 3; fs = f /@ Range[m]; Do[If[Equal @@  fs, AppendTo[s, n - m]]; fs = Rest @ AppendTo[fs, f[n]], {n, m + 1, 155000}]; s
SequencePosition[Table[Total[DigitCount[Divisors[n],2,1]],{n,160000}],{x_,x_,x_}][[All,1]] (* Harvey P. Dale, Feb 04 2023 *)

A332387 Numbers k such that k, k + 1 and k + 2 have the same number of divisors in Eisenstein integers.

Original entry on oeis.org

13448, 27848, 75774, 135400, 243338, 276123, 396950, 452823, 497575, 524823, 565674, 587575, 632224, 639848, 719223, 769316, 861123, 935799, 1060904, 1073875, 1153023, 1204312, 1308856, 1366624, 1413498, 1490599, 1555975, 1565223, 1601798, 1767424, 1902774, 1923295

Offset: 1

Amiram Eldar, Feb 10 2020

nonn

			13448 is a term since 13448, 13449 and 13450 each have 12 divisors in Eisenstein integers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005238, A319442, A332313, A332386, A332388.

f[p_, e_] := Switch[Mod[p, 3], 0, 2*e + 1, 1, (e + 1)^2, 2, e + 1]; eisNumDiv[1] = 1; eisNumDiv[n_] := Times @@ f @@@ FactorInteger[n]; Flatten[Position[Partition[ eisNumDiv /@ Range[10^6], 3, 1], {x_, x_, x_}]] (* after Harvey P. Dale at A005238 *)

A113467 Least k such that k, k+n and k+2n have the same number of divisors.

Original entry on oeis.org

33, 3, 119, 3, 77, 5, 8, 3, 77, 3, 35, 5, 8, 3, 187, 6, 21, 5, 8, 3, 145, 33, 39, 5, 8, 39, 8, 3, 33, 7, 15, 12, 189, 3, 28, 7, 21, 3, 55, 3, 33, 5, 8, 66, 209, 69, 35, 5, 8, 3, 115, 39, 141, 5, 51, 6, 8, 27, 15, 7, 21, 66, 95, 3, 40, 5, 27, 3, 8, 15, 35, 7, 69, 55, 287, 6, 65, 11, 8, 3, 24

Offset: 1

David Wasserman, Jan 08 2006

Third row of A113465.

			a(7) = 8 because 8, 15 and 22 each have 4 divisors.

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

Cf. A005238, A113458, A113465.

snd[n_]:=Module[{k=1},While[Length[Union[DivisorSigma[0,{k,k+n,k+2n}]]]>1, k++];k]; Array[snd,90] (* Harvey P. Dale, Aug 20 2017 *)

a(n) = {k  = 1; until ((numdiv(k) == numdiv(k+n)) && (numdiv(k) == numdiv(k+2*n)), k++); return (k);} \\ Michel Marcus, Jun 16 2013

cp's OEIS Frontend

A039665 Sets of 4 consecutive numbers with equal number of divisors.

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A169834 Numbers k such that d(k-1) = d(k) = d(k+1).

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A049053 Numbers k such that k through k+6 all have the same number of divisors.

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A049052 Numbers k such that k through k+5 all have the same number of divisors.

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A075032 Numbers n such that tau(n) <= tau(n+1) <= tau(n+2) where tau(n) = number of divisors of n.

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A292580 T(n,k) is the start of the first run of exactly k consecutive integers having exactly 2n divisors. Table read by rows.

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A332313 Numbers k such that k, k + 1 and k + 2 have the same number of divisors in Gaussian integers.

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Mathematica

A338453 Starts of runs of 3 consecutive numbers with the same total binary weight of their divisors (A093653).

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Mathematica

A332387 Numbers k such that k, k + 1 and k + 2 have the same number of divisors in Eisenstein integers.

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