cp's OEIS Frontend

A378253 Perfect powers p such that there are no other perfect powers between p and the least prime > p.

%I A378253 #8 Nov 26 2024 17:13:36
%S A378253 1,4,9,16,27,36,49,64,81,100,125,128,144,169,196,216,225,243,256,289,
%T A378253 324,343,361,400,441,484,512,529,576,625,676,729,784,841,900,961,1000,
%U A378253 1024,1089,1156,1225,1296,1331,1369,1444,1521,1600,1681,1728,1764,1849
%N A378253 Perfect powers p such that there are no other perfect powers between p and the least prime > p.
%C A378253 Perfect powers (A001597) are 1 and numbers with a proper integer root, complement A007916.
%C A378253 Each term is the greatest perfect power < prime(k) for some k.
%e A378253 The first number line below shows the perfect powers. The second shows each prime. To get a(n), we take the last perfect power in each interval between consecutive primes, omitting the cases where there are none.
%e A378253 -1-----4-------8-9------------16----------------25--27--------32------36----
%e A378253 ===2=3===5===7======11==13======17==19======23==========29==31==========37==
%t A378253 radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
%t A378253 Union[Table[NestWhile[#-1&,Prime[n],radQ[#]&],{n,1000}]]
%Y A378253 Union of A378035, restriction of A081676 to the primes.
%Y A378253 The opposite is A378250, union of A378249 (run-lengths A378251).
%Y A378253 A000040 lists the primes, differences A001223.
%Y A378253 A001597 lists the perfect powers, differences A053289.
%Y A378253 A007916 lists the non-perfect powers, differences A375706.
%Y A378253 A069623 counts perfect powers <= n.
%Y A378253 A076411 counts perfect powers < n.
%Y A378253 A080769 counts primes between perfect powers.
%Y A378253 A377283 ranks perfect powers between primes, differences A378356.
%Y A378253 A377432 counts perfect powers between primes, see A377434, A377436, A377466.
%Y A378253 Cf. A000015, A000961, A052410, A067871, A076412, A131605, A188951, A216765, A345531, A377057, A377468, A378355.
%K A378253 nonn
%O A378253 1,2
%A A378253 _Gus Wiseman_, Nov 26 2024