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Number of true prime powers whose binary order, ceiling(log_2(p^x)), is n.
2

%I #15 Oct 29 2020 21:58:54

%S 0,1,1,2,3,2,4,3,4,6,5,9,10,11,17,15,26,31,39,53,68,90,125,159,216,

%T 290,391,536,719,971,1329,1812,2477,3386,4626,6351,8729,11995,16459,

%U 22669,31259,43049,59388,82024,113275,156558,216560,299566,414821,574654

%N Number of true prime powers whose binary order, ceiling(log_2(p^x)), is n.

%e The 5 prime powers between 1025 and 2048 (inclusive) are 1331 = 11^3, 1369 = 37^2, 1681 = 41^2, 1849 = 43^2, and 2048 = 2^11.

%t t=Table[Length[Union[Flatten[Table[Table[Prime[w]^s, {w, 1, PrimePi[2^(n/s)]}], {s, 2, g+1}]]] ], {n, 1, 42}] Delete[t-RotateRight[t], 1]

%Y Cf. A029837, A036378-A036390.

%K nonn

%O 1,4

%A _Labos Elemer_

%E More terms from _Sean A. Irvine_, Oct 29 2020