Find smallest $n$ fulfilling requirement
For numbers $1$ to $2k \, (= 2000)$, the minimum size $n$ of a random set that will ensure, there is at least one pair of $a, 2a$ present in the set -
Basically it works out to $n \displaystyle \ge 1 + \sum \limits_{i \ge 0} \lceil \frac{k}{2^{2i}}\rceil \, (2^{2i} \le 2k$, even powers of $2$). I think we can write a closed form but will be approximation.
Details -
Any or all of the odd numbers can be randomly present. So we need at least $k+1$ numbers.
So $ n \ge 1001$.
Now if we choose any number of the form $\, 2(2m-1)$ where $(m \ge 1, \in \mathbb{Z+})$ to be in the set, the set will have at least one pair of $(a, 2a)$.
But if the randomly chosen number to be in the set is of the form $4(2m-1)$ instead of $2(2m-1)$, then we still do not have a pair of $a,2a$.
Numbers of the form $4(2m-1) = 250 \,$ ($\lceil \frac{k}{4} \rceil)$.
So we are covered if the randomly chosen numbers are of the form $2(2m-1), 4(2m-1), 8(2m-1)$ but not covered for $4^2(2m-1)$.
Numbers of the form $4^2(2m-1) = 63 \,$ ($\lceil \frac{k}{4^2} \rceil)$.
Similarly, numbers of the form $4^3(2m-1) = 16 \,$ ($\lceil \frac{k}{4^3} \rceil)$.
numbers of the form $4^4(2m-1) = 4 \,$ ($\lceil \frac{k}{4^4} \rceil)$.
numbers of the form $4^5(2m-1) = 1\,$ ($\lceil \frac{k}{4^5} \rceil)$.
So, $n \ge 1001 + 250 + 63 + 16 + 4 + 1 = 1335$ will ensure there is at least a pair of numbers of the form $a,2a$.
You can solve the problem via integer linear programming as follows. For $t\in T$, let binary decision variable $x_t$ indicate whether $t$ is selected. We want to maximize $\sum_t x_t$ subject to $x_t + x_{2t} \le 1$ for all $t\in\{1,\dots,k\}$. The maximum turns out to be $1334$, which means that $n=1334+1=1335$. One such optimal solution has $x_t=1$ for
$$t\in T\setminus\{2,6,8,10,14,18,22,24,26,30,32,34,38,40,42,46,50,54,56,58,62,66,70,72,74,78,82,
86,88,90,94,96,98,102,104,106,110,114,118,120,122,126,128,130,134,136,138,142,146,150,152,154,
158,160,162,166,168,170,174,178,182,184,186,190,194,198,200,202,206,210,214,216,218,222,224,226
,230,232,234,238,242,246,248,250,254,258,262,264,266,270,274,278,280,282,286,288,290,294,296,
298,302,306,310,312,314,318,322,326,328,330,334,338,342,344,346,350,352,354,358,360,362,366,370
,374,376,378,382,384,386,390,392,394,398,402,406,408,410,414,416,418,422,424,426,430,434,438,
440,442,446,450,454,456,458,462,466,470,472,474,478,480,482,486,488,490,494,498,501,502,503,504
,505,506,507,509,510,511,512,513,514,515,517,518,519,520,521,522,523,525,526,527,529,530,531,
533,534,535,536,537,538,539,541,542,543,544,545,546,547,549,550,551,552,553,554,555,557,558,559
,561,562,563,565,566,567,568,569,570,571,573,574,575,577,578,579,581,582,583,584,585,586,587,
589,590,591,593,594,595,597,598,599,600,601,602,603,605,606,607,608,609,610,611,613,614,615,616
,617,618,619,621,622,623,625,626,627,629,630,631,632,633,634,635,637,638,639,640,641,642,643,
645,646,647,648,649,650,651,653,654,655,657,658,659,661,662,663,664,665,666,667,669,670,671,672
,673,674,675,677,678,679,680,681,682,683,685,686,687,689,690,691,693,694,695,696,697,698,699,
701,702,703,705,706,707,709,710,711,712,713,714,715,717,718,719,721,722,723,725,726,727,728,729
,730,731,733,734,735,736,737,738,739,741,742,743,744,745,746,747,749,750,751,753,754,755,757,
758,759,760,761,762,763,765,766,767,769,770,771,773,774,775,776,777,778,779,781,782,783,785,786
,787,789,790,791,792,793,794,795,797,798,799,800,801,802,803,805,806,807,808,809,810,811,813,
814,815,817,818,819,821,822,823,824,825,826,827,829,830,831,833,834,835,837,838,839,840,841,842
,843,845,846,847,849,850,851,853,854,855,856,857,858,859,861,862,863,864,865,866,867,869,870,
871,872,873,874,875,877,878,879,881,882,883,885,886,887,888,889,890,891,893,894,895,896,897,898
,899,901,902,903,904,905,906,907,909,910,911,913,914,915,917,918,919,920,921,922,923,925,926,
927,928,929,930,931,933,934,935,936,937,938,939,941,942,943,945,946,947,949,950,951,952,953,954
,955,957,958,959,961,962,963,965,966,967,968,969,970,971,973,974,975,977,978,979,981,982,983,
984,985,986,987,989,990,991,992,993,994,995,997,998,999,1000,1016,1032,1048,1056,1064,1080,1096
,1112,1120,1128,1144,1152,1160,1176,1184,1192,1208,1224,1240,1248,1256,1272,1288,1304,1312,1320
,1336,1352,1368,1376,1384,1400,1408,1416,1432,1440,1448,1464,1480,1496,1504,1512,1528,1536,1544
,1560,1568,1576,1592,1608,1624,1632,1640,1656,1664,1672,1688,1696,1704,1720,1736,1752,1760,1768
,1784,1800,1816,1824,1832,1848,1864,1880,1888,1896,1912,1920,1928,1944,1952,1960,1976,1992\}$$
The linear programming relaxation also has optimal objective value $1334$, and the dual variables provide a certificate of optimality in the form of $666$ disjoint pairs $(t,2t)$:
$$P=\{(1,2),(3,6),(4,8),(5,10),(7,14),(9,18),(11,22),(12
,24),(13,26),(15,30),(16,32),(17,34),(19,38),(20,40),(21,42),(23,46),(25,50),(27,54),(28,56),(
29,58),(31,62),(33,66),(35,70),(36,72),(37,74),(39,78),(41,82),(43,86),(44,88),(45,90),(47,94),
(48,96),(49,98),(51,102),(52,104),(53,106),(55,110),(57,114),(59,118),(60,120),(61,122),(63,126
),(64,128),(65,130),(67,134),(68,136),(71,142),(73,146),(75,150),(77,154),(79,158),(80,160),(85
,170),(87,174),(91,182),(92,184),(93,186),(95,190),(99,198),(103,206),(105,210),(107,214),(108,
216),(111,222),(112,224),(113,226),(119,238),(127,254),(129,258),(131,262),(132,264),(133,266),
(135,270),(137,274),(138,276),(139,278),(140,280),(141,282),(143,286),(144,288),(145,290),(147,
294),(148,296),(149,298),(151,302),(152,304),(153,306),(155,310),(156,312),(157,314),(159,318),
(161,322),(162,324),(163,326),(164,328),(165,330),(166,332),(167,334),(168,336),(169,338),(171,
342),(172,344),(173,346),(175,350),(176,352),(177,354),(178,356),(179,358),(180,360),(181,362),
(183,366),(185,370),(187,374),(188,376),(189,378),(191,382),(192,384),(193,386),(194,388),(195,
390),(196,392),(197,394),(199,398),(200,400),(201,402),(202,404),(203,406),(204,408),(205,410),
(207,414),(208,416),(209,418),(211,422),(212,424),(213,426),(215,430),(217,434),(218,436),(219,
438),(220,440),(221,442),(223,446),(225,450),(227,454),(228,456),(229,458),(230,460),(231,462),
(232,464),(233,466),(234,468),(235,470),(236,472),(237,474),(239,478),(240,480),(241,482),(242,
484),(243,486),(244,488),(245,490),(246,492),(247,494),(248,496),(249,498),(250,500),(251,502),
(256,512),(260,520),(269,538),(272,544),(275,550),(283,566),(285,570),(287,574),(289,578),(292,
584),(297,594),(299,598),(301,602),(305,610),(307,614),(309,618),(311,622),(316,632),(317,634),
(320,640),(321,642),(323,646),(329,658),(337,674),(339,678),(340,680),(341,682),(343,686),(345,
690),(348,696),(353,706),(363,726),(365,730),(367,734),(368,736),(369,738),(372,744),(373,746),
(375,750),(377,754),(380,760),(381,762),(389,778),(391,782),(393,786),(395,790),(396,792),(401,
802),(405,810),(411,822),(412,824),(417,834),(425,850),(429,858),(431,862),(432,864),(433,866),
(441,882),(444,888),(448,896),(449,898),(453,906),(455,910),(461,922),(463,926),(465,930),(469,
938),(471,942),(476,952),(477,954),(479,958),(481,962),(485,970),(495,990),(499,998),(501,1002)
,(503,1006),(504,1008),(505,1010),(506,1012),(507,1014),(508,1016),(509,1018),(510,1020),(511,
1022),(513,1026),(514,1028),(515,1030),(516,1032),(517,1034),(518,1036),(519,1038),(521,1042),(
522,1044),(523,1046),(524,1048),(525,1050),(526,1052),(527,1054),(528,1056),(529,1058),(530,
1060),(531,1062),(532,1064),(533,1066),(534,1068),(535,1070),(536,1072),(537,1074),(539,1078),(
540,1080),(541,1082),(542,1084),(543,1086),(545,1090),(546,1092),(547,1094),(548,1096),(549,
1098),(551,1102),(552,1104),(553,1106),(554,1108),(555,1110),(556,1112),(557,1114),(558,1116),(
559,1118),(560,1120),(561,1122),(562,1124),(563,1126),(564,1128),(565,1130),(567,1134),(568,
1136),(569,1138),(571,1142),(572,1144),(573,1146),(575,1150),(576,1152),(577,1154),(579,1158),(
580,1160),(581,1162),(582,1164),(583,1166),(585,1170),(586,1172),(587,1174),(588,1176),(589,
1178),(590,1180),(591,1182),(592,1184),(593,1186),(595,1190),(596,1192),(597,1194),(599,1198),(
600,1200),(601,1202),(603,1206),(604,1208),(605,1210),(606,1212),(607,1214),(608,1216),(609,
1218),(611,1222),(612,1224),(613,1226),(615,1230),(616,1232),(617,1234),(619,1238),(620,1240),(
621,1242),(623,1246),(624,1248),(625,1250),(626,1252),(627,1254),(628,1256),(629,1258),(630,
1260),(631,1262),(633,1266),(635,1270),(636,1272),(637,1274),(638,1276),(639,1278),(641,1282),(
643,1286),(644,1288),(645,1290),(647,1294),(648,1296),(649,1298),(650,1300),(651,1302),(652,
1304),(653,1306),(654,1308),(655,1310),(656,1312),(657,1314),(659,1318),(660,1320),(661,1322),(
662,1324),(663,1326),(664,1328),(665,1330),(666,1332),(667,1334),(668,1336),(669,1338),(670,
1340),(671,1342),(672,1344),(673,1346),(675,1350),(676,1352),(677,1354),(679,1358),(681,1362),(
683,1366),(684,1368),(685,1370),(687,1374),(688,1376),(689,1378),(691,1382),(692,1384),(693,
1386),(694,1388),(695,1390),(697,1394),(698,1396),(699,1398),(700,1400),(701,1402),(702,1404),(
703,1406),(704,1408),(705,1410),(707,1414),(708,1416),(709,1418),(710,1420),(711,1422),(712,
1424),(713,1426),(714,1428),(715,1430),(716,1432),(717,1434),(718,1436),(719,1438),(720,1440),(
721,1442),(722,1444),(723,1446),(724,1448),(725,1450),(727,1454),(728,1456),(729,1458),(731,
1462),(732,1464),(733,1466),(735,1470),(737,1474),(739,1478),(740,1480),(741,1482),(742,1484),(
743,1486),(745,1490),(747,1494),(748,1496),(749,1498),(751,1502),(752,1504),(753,1506),(755,
1510),(756,1512),(757,1514),(758,1516),(759,1518),(761,1522),(763,1526),(764,1528),(765,1530),(
766,1532),(767,1534),(768,1536),(769,1538),(770,1540),(771,1542),(772,1544),(773,1546),(774,
1548),(775,1550),(776,1552),(777,1554),(779,1558),(780,1560),(781,1562),(783,1566),(784,1568),(
785,1570),(787,1574),(788,1576),(789,1578),(791,1582),(793,1586),(794,1588),(795,1590),(796,
1592),(797,1594),(798,1596),(799,1598),(800,1600),(801,1602),(803,1606),(804,1608),(805,1610),(
806,1612),(807,1614),(808,1616),(809,1618),(811,1622),(812,1624),(813,1626),(814,1628),(815,
1630),(816,1632),(817,1634),(818,1636),(819,1638),(820,1640),(821,1642),(823,1646),(825,1650),(
826,1652),(827,1654),(828,1656),(829,1658),(830,1660),(831,1662),(832,1664),(833,1666),(835,
1670),(836,1672),(837,1674),(838,1676),(839,1678),(840,1680),(841,1682),(842,1684),(843,1686),(
844,1688),(845,1690),(846,1692),(847,1694),(848,1696),(849,1698),(851,1702),(852,1704),(853,
1706),(854,1708),(855,1710),(856,1712),(857,1714),(859,1718),(860,1720),(861,1722),(863,1726),(
865,1730),(867,1734),(868,1736),(869,1738),(870,1740),(871,1742),(872,1744),(873,1746),(874,
1748),(875,1750),(876,1752),(877,1754),(878,1756),(879,1758),(880,1760),(881,1762),(883,1766),(
884,1768),(885,1770),(886,1772),(887,1774),(889,1778),(890,1780),(891,1782),(892,1784),(893,
1786),(894,1788),(895,1790),(897,1794),(899,1798),(900,1800),(901,1802),(902,1804),(903,1806),(
904,1808),(905,1810),(907,1814),(908,1816),(909,1818),(911,1822),(912,1824),(913,1826),(914,
1828),(915,1830),(916,1832),(917,1834),(918,1836),(919,1838),(920,1840),(921,1842),(923,1846),(
924,1848),(925,1850),(927,1854),(928,1856),(929,1858),(931,1862),(932,1864),(933,1866),(934,
1868),(935,1870),(936,1872),(937,1874),(939,1878),(940,1880),(941,1882),(943,1886),(944,1888),(
945,1890),(946,1892),(947,1894),(948,1896),(949,1898),(950,1900),(951,1902),(953,1906),(955,
1910),(956,1912),(957,1914),(959,1918),(960,1920),(961,1922),(963,1926),(964,1928),(965,1930),(
966,1932),(967,1934),(968,1936),(969,1938),(971,1942),(972,1944),(973,1946),(974,1948),(975,
1950),(976,1952),(977,1954),(978,1956),(979,1958),(980,1960),(981,1962),(982,1964),(983,1966),(
984,1968),(985,1970),(986,1972),(987,1974),(988,1976),(989,1978),(991,1982),(992,1984),(993,
1986),(994,1988),(995,1990),(996,1992),(997,1994),(999,1998),(1000,2000)
\}$$
Because at most one in each pair can be selected, this set $P$ of disjoint pairs proves that $$\sum_{t\in T} x_t \le |P| + (|T| - 2|P|) = |T| - |P| = 2000 - 666 = 1334.$$