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de Polignac n.  and  powerful n.

First numbers that belong to both sets:

1, 40401, 62001, 96721, 121801, 192721, 205379, 226981, 326041, 389017, 410881, 555025, 660969, 683929, 704099, 757687, 772641, 786769, 822649, 1119371, 1343281, 1390041, 1530169, 1685159, 1729063, 1739761, 1827847, 1885129, 1923769, 1962801, 2237411, 2283121, 2461019, 2481997, 2544025, 2803221, 2913849, 3195731, 3207681, 3214849, 3227287, 3352561, 3396649, 3613801, 3693157, 3721925, 3738779, 3775249, 3853369, 4060225, 4289041, 4330747, 4338889, 4397409, 4490161, 4756761, 4800481, 4879681, 5058001, 5175625, 5527201, 5736025, 5745609, 5851561, 6295081, 6487209, 6548481, 6627103, 6709571, 6967871, 7017201, 7092899, 7155625, 7230721, 7241481, 7252249, 7789681, 8003241, 8173881, 8185321, 8323225, 8334769, 8804429, 9113167, 9126441, 9603801, 9620347, 9840769, 10063571, 10182481, 10400625, 10542851, 10543009, 10804369, 10896601, 10990063, 11089567, 11108889, 11363641, 11785489, 12159169, 12313081, 12523379, 12766329, 13203125, 13651919, 13711477, 14010049, 14120579, 14478025, 14538969, 14706125, 15069223, 15077689, 15151531, 15238619, 15531481, 15636267, 17480761, 17674447, 18285733, 19061833, 19193161, 19439281, 19651489, 20493729, 20671573, 20729809, 20912329, 21945125, 22212369, 22370117, 22439169, 22553001, 22648081, 22665187, 22953681, 23008783, 23011209, 23876179, 24039409, 24117921, 24235929, 24453025, 24730729, 24819823, 24880481, 24981539, 25010001, 25090081, 25153453, 25371369, 25411681, 25816561, 25999801, 26106125, 26623323, 26876903, 27217619, 27615025, 27825625, 28252567, 28804689, 28934443, 28976689, 29452329, 29560969, 29646125, 29876525, 30283009, 30415225, 30680521, 31775769, 31878125, 32001649, 32364721, 32433025, 32638369, 32844361, 32959081, 33175303, 33466225, 33744481, 34140649, 34234201, 34468641, 34869025, 35414401, 35772361, 35784847, 36180225, 36397089, 36469521, 36566209, 36784225, 37027225, 37491129, 38014691, 38954167, 39530583, 39884383, 39955041, 40564161, 40998409, 41254929, 41589601, 41770369, 42601729, 43151761, 44555625, 44635761, 45095261, 45576001, 45846441, 46543739, 47018449, 47210641, 47265625, 47844889, 47895125, 48233025, 48539089, 48762289, 49463089, 49556275, 49578419, 49632025, 50093797, 50314327, 50537881, 50822641, 51903103, 51912025, 51998521, 52171729, 52345225, 52708931, 54059887, 54243225, 54656449, 55398249, 55487601, 55845729, 55875625, 56115081, 56385081, 56505289, 57377383, 57562569, 57714409, 57927321, 58415449, 58599025, 59706529, 60027107, 60512841, 60855601, 61074225, 61333847, 61512649, 61732449, 62932489, 63409369, 63984001, 64481201, 65399569, 66447513, 67354849, 67617729, 68873401, 69343957, 69872881, 70509609, 72131049, 72199009, 72353763, 73017025, 73359225, 73599241, 73776725.

Some larger examples :