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gapful n.  and  Poulet n.

First numbers that belong to both sets:

341, 561, 1729, 1905, 13981, 18705, 39865, 41041, 101101, 113201, 129921, 149281, 157641, 176149, 348161, 399001, 410041, 625921, 825265, 898705, 1004653, 1023121, 1139281, 1157689, 1246785, 1275681, 1472505, 1489665, 1507561, 1533961, 1569457, 1608465, 1773289, 1815465, 1827001, 1896961, 1907851, 1937881, 3020361, 3146221, 3181465, 3435565, 3726541, 3916261, 4154161, 4229601, 4363261, 4806061, 4984001, 5133201, 5560809, 6189121, 6309901, 7516153, 8322945, 9890881, 10226161, 10513261, 10837321, 10877581, 10956673, 11075857, 11205601, 11433301, 13333441, 13696033, 13757653, 13823601, 14870801, 14898631, 15124969, 15716041, 16053193, 16263105, 17020201, 17450569, 17698241, 18137505, 18162001, 18468901, 18779761, 18900973, 19523505, 27218269, 27336673, 30069721, 30653245, 30996001, 31146661, 33302401, 33848311, 35571601, 35932441, 36168265, 36507801, 37727341, 38342071, 38584801, 40928701, 41396921, 41424801, 41541241, 42490801, 45393601, 46104697, 48321001, 48650641, 49472281, 57421219, 60957361, 61198921, 62490901, 64644385, 67371265, 68830021, 69776785, 75151441, 75765313, 77477401, 78073801, 81722145, 84234745, 84311569, 93614521, 97420141, 98124481, 100302301, 100427041, 102757501, 103065945, 104404861, 105085981, 105517621, 105569101, 105869401, 106041937, 108145741, 109245585, 110671341, 110851741, 114362281, 115756201, 116397345, 120570409, 121422301, 122569993, 122649901, 125707681, 125739361, 128039241, 128626219, 129255841, 129604321, 131585421, 132841369, 136446465, 137897881, 138652921, 139489251, 139592101, 140285377, 142825033, 143258689, 144093313, 144200001, 145082201, 150886681, 151530401, 151833561, 153927961, 153958321, 154364221, 155172165, 155255881, 156858801, 161030001, 161913961, 166724041, 169057801, 171804061, 172295761, 172430401, 173085121, 173928601, 181234801, 181299811, 181397161, 181449073, 182082901, 182356993, 184527101, 187188001, 187823141, 188082301, 188481601, 188689501, 190667851, 191752093, 192112921, 192736965, 193009861, 193708801, 194675041, 197200201, 197781221, 269953489, 296639869, 300614161, 301030801, 301068001, 302751505, 306703801, 306871201, 310302901, 311388337, 317657341, 321524281, 321602401, 321831181, 325546585, 329379505, 332633665, 334849321, 335429641, 339396401, 348140881, 353987481, 357339745, 358466361, 358940737, 360067201, 364590721, 364992481, 367804801, 369269551, 370851481, 370874245, 377199661, 382536001, 385924105, 390922741, 391938301, 419394781, 419520241, 421121701, 421335721, 434932961, 445448641, 452877841, 454934401, 455106601, 464021641, 471441001, 490806121, 496050841, 496803553, 498195961, 556418721, 571614681, 580565233, 602593441, 605033065, 606057985, 607677181, 608442121, 611146861, 611397865, 612347905, 613457065, 619268401, 629350345, 629470465, 639925441, 645986401, 647101585, 648328801, 652684201, 658126621, 672389641, 676280221.

Some larger examples :