A075040 -id:A075040 - cp's OEIS frontend

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

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.

A306879 Smallest number m such that m, m+1, and m+2 all have exactly 2p divisors, where p = prime(n).

Original entry on oeis.org

33, 242, 7939375, 76571890623, 104228508212890623, 1489106237081787109375, 273062471666259918212890623, 804505911103256259918212890623, 490685203356467392256259918212890623, 6794675247932944436619977392256259918212890623, 329757106427071213106619977392256259918212890623

Offset: 1

Chai Wah Wu, Mar 14 2019

nonn

a(4) was incorrect in "Some new results on consecutive equidivisible integers".

			33, 34, 35 all have exactly 2*prime(1) = 4 divisors, and 33 is the smallest number with this property, so a(1) = 33.

Chai Wah Wu, Table of n, a(n) for n = 1..50
Vasilii A. Dziubenko, Vladimir A. Letsko, Consecutive positive integers with the same number of divisors, arXiv:1811.05127 [math.NT], 2018.
Vladimir A. Letsko, Some new results on consecutive equidivisible integers, arXiv:1510.07081 [math.NT], 2015.

Cf. A274639.

A subsequence of A075040.

A274639 Erroneous version of A306879.

Original entry on oeis.org

33, 242, 7939375, 3388031791, 104228508212890623, 1489106237081787109375, 273062471666259918212890623, 804505911103256259918212890623, 490685203356467392256259918212890623, 6794675247932944436619977392256259918212890623, 329757106427071213106619977392256259918212890623

Offset: 1

N. J. A. Sloane, Jul 09 2016

dead

A published but incorrect sequence. The OEIS policy is to include such sequences together with a pointer to the correct entry.

a(4) was incorrect in "Some new results on consecutive equidivisible integers".

			33, 34, 35 all have exactly 2*prime(1) = 4 divisors, and 33 is the smallest number with this property, so a(1) = 33.

Table of n, a(n) for n = 1..11
Vasilii A. Dziubenko, Vladimir A. Letsko, Consecutive positive integers with the same number of divisors, arXiv:1811.05127 [math.NT], 2018.
Vladimir A. Letsko, Some new results on consecutive equidivisible integers, arXiv:1510.07081 [math.NT], 2015.

Cf. A306879.

Supposed to be subsequence of A075040.

cp's OEIS Frontend

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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A306879 Smallest number m such that m, m+1, and m+2 all have exactly 2p divisors, where p = prime(n).

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A274639 Erroneous version of A306879.

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