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%I #200 Jun 26 2023 14:31:10
%S 5,2,6,14,33,12,44,603,242,10093613546512321,24,104,230,3655,11605,
%T 28374,171893,48,2511,7939375,60,735,1274,19940,204323,368431323,
%U 155385466971,18652995711772,15724736975643,2973879756088065948,9887353188984012120346
%N T(n,k) is the start of the first run of exactly k consecutive integers having exactly 2n divisors. Table read by rows.
%C The number of terms in row n is A119479(2n).
%C 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).
%C 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).
%H Hugo van der Sanden, <a href="/A292580/b292580.txt">Table of n, a(n) for n = 1..32</a>
%H Ivo Düntsch and Roger B. Eggleton, <a href="http://www.cosc.brocku.ca/~duentsch/papers/equidiv.html">Equidivisible consecutive integers</a>, 1989.
%H Vladimir A. Letsko, <a href="http://arxiv.org/abs/1510.07081">Some new results on consecutive equidivisible integers</a>, arXiv:1510.07081 [math.NT], 2015.
%H Vladimir A. Letsko, <a href="http://www-old.fizmat.vspu.ru/lib/exe/fetch.php?media=marathon:consec_table_27-06-17.pdf">Table of a(n) for all even n such that exact value of a(n) is proved</a>, 2017.
%H Carlos Rivera, <a href="http://www.primepuzzles.net/problems/prob_020.htm">Problem 20: k consecutive numbers with the same number of divisors</a>, The Prime Puzzles and Problems Connection.
%H Hugo van der Sanden, <a href="https://github.com/hvds/divrep/wiki/D(12,11)">calculation of T(6,11)</a>.
%H Hugo van der Sanden, <a href="https://github.com/hvds/divrep/wiki/D(12,12)">calculation of T(6,12)</a>.
%H Hugo van der Sanden, <a href="/A292580/a292580_11.txt">A-file with known values and bounds up to T(50,7)</a>
%F T(n,2) = A075036(n). - _Jon E. Schoenfield_, Sep 23 2017
%e 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);
%e 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);
%e 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.
%e Table begins:
%e n T(n,1), T(n,2), ...
%e == ========================================================
%e 1 5, 2;
%e 2 6, 14, 33;
%e 3 12, 44, 603, 242, 10093613546512321;
%e 4 24, 104, 230, 3655, 11605, 28374, 171893;
%e 5 48, 2511, 7939375;
%e 6 60, 735, 1274, 19940, 204323, 368431323, 155385466971, 18652995711772, 15724736975643, 2973879756088065948, 9887353188984012120346, 120402988681658048433948, T(6,13), ...;
%e 7 192, 29888, 76571890623;
%e 8 120, 2295, 8294, 153543, 178086, 5852870, 17476613;
%e 9 180, 6075, 959075, 66251139635486389922, T(9,5);
%e 10 240, 5264, 248750, 31805261872, 1428502133048749, 8384279951009420621, 189725682777797295066519373;
%e 11 3072, 2200933376, 104228508212890623;
%e 12 360, 5984, 72224, 2919123, 15537948, 973277147, 33815574876, 1043710445721, 2197379769820, 2642166652554075, 17707503256664346, T(12,12), ...;
%e 13 12288, 689278976, 1489106237081787109375;
%e 14 960, 156735, 23513890624, 4094170438109373, 55644509293039461218749, 4230767238315793911295500109374, 273404501868270838132985214432619890621;
%e 15 720, 180224, 145705879375, 10868740069638250502059754282498, T(15,5);
%e 16 840, 21735, 318680, 6800934, 57645182, 1194435205, 14492398389;
%e ...
%Y Cf. A000005, A006558, A072507, A119479.
%Y Cf. A005237, A005238, A006601, A049051, A049052, A049053.
%Y Cf. A003680, A075036, A075040.
%K nonn,tabf,more
%O 1,1
%A _Jon E. Schoenfield_, Sep 19 2017
%E a(1)-a(25) from Düntsch and Eggleton (1989) with corrections by _Jon E. Schoenfield_, Sep 19 2017
%E a(26)-a(27) from _Giovanni Resta_, Sep 20 2017
%E a(28)-a(29) from _Hugo van der Sanden_, Jan 12 2022
%E a(30) from _Hugo van der Sanden_, Sep 03 2022
%E 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.
%E 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.
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