A005237 -id:A005237 - cp's OEIS frontend

A344312 Number k such that k and k+1 have the same number of exponential divisors (A049419).

1, 2, 5, 6, 8, 10, 13, 14, 21, 22, 24, 27, 29, 30, 33, 34, 37, 38, 41, 42, 44, 46, 49, 57, 58, 61, 65, 66, 69, 70, 73, 75, 77, 78, 80, 82, 85, 86, 93, 94, 98, 101, 102, 105, 106, 109, 110, 113, 114, 116, 118, 120, 122, 124, 125, 129, 130, 133, 135, 137, 138, 141

Offset: 1

Amiram Eldar, May 14 2021

nonn

			1 is a term since A049419(1) = A049419(2) = 1.
8 is a term since A049419(8) = A049419(9) = 2.

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

Cf. A049419.

Similar sequences: A005237, A006049, A343819, A344313, A344314.

f[p_, e_] := DivisorSigma[0, e]; ed[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[200], ed[#] == ed[# + 1] &]

A344313 Number k such that k and k+1 have the same number of bi-unitary divisors (A286324).

Original entry on oeis.org

2, 3, 4, 14, 15, 20, 21, 26, 27, 33, 34, 35, 38, 44, 45, 50, 51, 57, 62, 68, 74, 75, 76, 81, 85, 86, 91, 92, 93, 94, 98, 99, 104, 115, 116, 117, 118, 122, 123, 124, 133, 135, 141, 142, 145, 146, 147, 158, 171, 177, 187, 189, 201, 202, 205, 206, 212, 213, 214

Offset: 1

Amiram Eldar, May 14 2021

nonn

			2 is a term since A286324(2) = A286324(3) = 2.
14 is a term since A286324(14) = A286324(15) = 4.

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

Cf. A286324, A293183, A304463.

Similar sequences: A005237, A006049, A343819, A344312, A344314.

f[p_, e_] := If[OddQ[e], e + 1, e]; bd[1] = 1; bd[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[200], bd[#] == bd[# + 1] &]

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 *)

A215197 Numbers k such that k and k + 1 are both of the form p*q^4 where p and q are distinct primes.

Original entry on oeis.org

2511, 7856, 10287, 15471, 15632, 18063, 20816, 28592, 36368, 40816, 54512, 75248, 88047, 93231, 101168, 126927, 134703, 160624, 163376, 170991, 178767, 210032, 215216, 217808, 220624, 254096, 256527, 274671, 280624, 292976, 334448, 347408, 443151, 482192

Offset: 1

Michel Lagneau, Aug 05 2012

nonn

The smaller of adjacent terms in A178739. - R. J. Mathar, Aug 08 2012

These are numbers n such that n and n+1 both have 10 divisors. Proof: clearly n and n+1 cannot both be of the form p^9, so for contradiction assume either n and n+1 is of the form p*q^4 and the other is of the form r^9 where p, q, and r are prime. So p*q^4 is either r^9 - 1 = (r-1)(r^2+r+1)(r^6+r^3+1) or r^9 + 1 = (r+1)(r^2-r+1)(r^6-r^3+1). But these factors are relatively prime and so cannot represent p*q^4 unless one or more factors are units. But this does not happen for r > 2, and the case r = 2 does not work since neither 511 not 513 is of the form p*q^4. - Charles R Greathouse IV, Jun 19 2016

			2511 is a member as 2511 = 31*3^4 and 2512 = 157*2^4.

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

Intersection of A005237 and A030628.

Cf. A074172, A215173.

with(numtheory):for n from 3 to 500000 do:x:=factorset(n):y:=factorset(n+1):n1:=nops(x):n2:=nops(y):if n1=2 and n2=2 then xx1:=x[1]*x[2]^4 : xx2:=x[2]*x[1]^4:yy1:=y[1]*y[2]^4: yy2:=y[2]*y[1]^4:if (xx1=n or xx2=n) and (yy1=n+1 or yy2=n+1) then printf("%a, ", n):else fi:fi:od:

lst={}; Do[f1=FactorInteger[n]; If[Sort[Transpose[f1][[2]]]=={1, 4}, f2=FactorInteger[n+1]; If[Sort[Transpose[f2][[2]]]=={1, 4}, AppendTo[lst, n]]], {n, 3, 55000}]; lst
(* First run program for A178739 *) Select[A178739, MemberQ[A178739, # + 1] &] (* Alonso del Arte, Aug 05 2012 *)

is(n)=numdiv(n)==10 && numdiv(n+1)==10 \\ Charles R Greathouse IV, Jun 19 2016

is(n)=vecsort(factor(n)[,2])==[1,4]~ && vecsort(factor(n+1)[,2])==[1,4]~ \\ Charles R Greathouse IV, Jun 19 2016

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.

A332386 Numbers k such that k and k + 1 have the same number of divisors in Eisenstein integers.

Original entry on oeis.org

3, 7, 32, 50, 68, 174, 184, 200, 212, 219, 247, 291, 328, 343, 368, 376, 435, 472, 495, 543, 579, 608, 644, 679, 712, 716, 723, 788, 795, 849, 860, 871, 874, 904, 932, 939, 1011, 1015, 1058, 1074, 1076, 1159, 1184, 1220, 1227, 1336, 1359, 1384, 1436, 1495, 1515

Offset: 1

Amiram Eldar, Feb 10 2020

nonn

			3 is a term since 3 and 4 both have 3 divisors in Eisenstein integers.

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

Cf. A005237, A319442, A332312, A332387, 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]; SequencePosition[eisNumDiv /@ Range[1520], {x_, x_}][[All, 1]] (* after Harvey P. Dale at A005237 *)

A358817 Numbers k such that A046660(k) = A046660(k+1).

Original entry on oeis.org

1, 2, 5, 6, 10, 13, 14, 21, 22, 29, 30, 33, 34, 37, 38, 41, 42, 44, 46, 49, 57, 58, 61, 65, 66, 69, 70, 73, 75, 77, 78, 80, 82, 85, 86, 93, 94, 98, 101, 102, 105, 106, 109, 110, 113, 114, 116, 118, 122, 129, 130, 133, 135, 137, 138, 141, 142, 145, 147, 154, 157

Offset: 1

Amiram Eldar, Dec 02 2022

nonn

First differs from its subsequence A007674 at n=18.

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 5, 38, 369, 3655, 36477, 364482, 3644923, 36449447, 364494215, 3644931537, ... . Apparently, the asymptotic density of this sequence exists and equals 0.36449... .

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

Cf. A046660.
Subsequences: A007674, A052213, A085651, A358818.
Similar sequences: A002961, A005237, A006049, A045920.

seq[kmax_] := Module[{s = {}, e1 = 0, e2}, Do[e2 = PrimeOmega[k] - PrimeNu[k]; If[e1 == e2, AppendTo[s, k - 1]]; e1 = e2, {k, 2, kmax}]; s]; seq[160]

e(n) = {my(f = factor(n)); bigomega(f) - omega(f)};
lista(nmax) = {my(e1 = e(1), e2); for(n=2, nmax, e2=e(n); if(e1 == e2, print1(n-1,", ")); e1 = e2);}

A054005 Sum of divisors of k such that k and k+1 have the same number and sum of divisors.

Original entry on oeis.org

24, 2160, 2640, 4320, 51840, 65280, 115200, 138240, 194400, 186048, 276480, 483840, 622080, 700416, 950400, 984960, 1118880, 1128960, 1612800, 2661120, 3937248, 3617280, 5019840, 6128640, 5806080, 7375680, 8467200, 11583936

Offset: 1

Asher Auel, Jan 12 2000

nonn

			See example in A054004.

Amiram Eldar, Table of n, a(n) for n = 1..1831

Cf. A000005, A000203, A002961, A005237, A053249, A054004, A054006, A054007.

Select[Partition[Table[{n,DivisorSigma[0,n],DivisorSigma[1,n]},{n,116*10^5}],2,1],#[[1,2]]== #[[2,2]] && #[[1,3]]==#[[2,3]]&][[All,1,3]] (* Harvey P. Dale, May 16 2023 *)

a(n) = sigma(A054004(n)).

More terms from Jud McCranie, Oct 15 2000

Definition clarified by Harvey P. Dale, May 16 2023

A054006 Number of divisors of k and k+1 which have the same number and sum of divisors.

Original entry on oeis.org

4, 8, 8, 8, 8, 8, 16, 16, 16, 8, 16, 16, 16, 16, 16, 16, 16, 16, 32, 16, 16, 16, 16, 32, 16, 16, 16, 24, 16, 16, 16, 32, 32, 32, 16, 16, 32, 16, 32, 16, 16, 32, 32, 16, 32, 16, 16, 16, 16, 16, 32, 32, 16, 32, 16, 16, 64, 32, 16, 32, 16, 32, 16, 64, 32, 32, 16, 32, 32, 32, 32

Offset: 1

Asher Auel, Jan 12 2000

nonn

			See example in A054004.

Amiram Eldar, Table of n, a(n) for n = 1..1831 (calculated using the b-file at A054004)

Cf. A000005, A000203, A002961, A005237, A053249, A054004, A054005, A054007.

Select[Partition[Array[DivisorSigma[{0, 1}, #] &, 10^6], 2, 1], SameQ @@ # &][[All, 1, 1]] (* Michael De Vlieger, Nov 21 2019 *)

a(n) = tau(A054004(n)).

More terms from Jud McCranie, Oct 15 2000

A054007 Numbers k such that k and k+1 have the same sum but an unequal number of divisors.

Original entry on oeis.org

206, 957, 1364, 2974, 4364, 14841, 18873, 19358, 20145, 24957, 36566, 56564, 74918, 79826, 79833, 92685, 111506, 116937, 138237, 147454, 161001, 162602, 174717, 190773, 193893, 201597, 230390, 274533, 347738, 416577, 422073, 430137

Offset: 1

Asher Auel, Jan 12 2000

nonn

			The divisors of 206 are 1, 2, 103, 206, so tau(206) = 4 and sigma(206) = 312; the divisors of 207 are 1, 3, 9, 23, 69, 207, so tau(207) = 6 and sigma(207) = 312. Hence, the integer 206 belongs to this sequence. - _Bernard Schott_, Oct 18 2019

Amiram Eldar, Table of n, a(n) for n = 1..8304 (terms below 10^13, calculated from the b-file at A002961)

Cf. A000005, A000203, A002961, A005237, A053249, A054004, A054005, A054006.

Select[Range[100000], DivisorSigma[0, #] != DivisorSigma[0, # + 1] && DivisorSigma[1, #] == DivisorSigma[1, # + 1] &] (* Jayanta Basu, Mar 20 2013 *)

Members of A002961 which are not members of A054004

More terms from Jud McCranie, Oct 15 2000

cp's OEIS Frontend

A344312 Number k such that k and k+1 have the same number of exponential divisors (A049419).

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A344313 Number k such that k and k+1 have the same number of bi-unitary divisors (A286324).

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

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A215197 Numbers k such that k and k + 1 are both of the form p*q^4 where p and q are distinct primes.

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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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A332386 Numbers k such that k and k + 1 have the same number of divisors in Eisenstein integers.

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A358817 Numbers k such that A046660(k) = A046660(k+1).

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A054005 Sum of divisors of k such that k and k+1 have the same number and sum of divisors.

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