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A378249 Least perfect power > prime(n).

%I A378249 #13 Dec 19 2024 10:25:06
%S A378249 4,4,8,8,16,16,25,25,25,32,32,49,49,49,49,64,64,64,81,81,81,81,100,
%T A378249 100,100,121,121,121,121,121,128,144,144,144,169,169,169,169,169,196,
%U A378249 196,196,196,196,216,216,216,225,243,243,243,243,243,256,289,289,289
%N A378249 Least perfect power > prime(n).
%C A378249 Perfect-powers (A001597) are numbers with a proper integer root, complement A007916.
%C A378249 Which terms appear only once? Just 128, 225, 256, 64009, 1295044?
%e A378249 The first number line below shows the perfect powers. The second shows each prime.
%e A378249 -1-----4-------8-9------------16----------------25--27--------32------36------------------------49--
%e A378249 ===2=3===5===7======11==13======17==19======23==========29==31==========37======41==43======47======
%t A378249 radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
%t A378249 Table[NestWhile[#+1&,Prime[n],radQ[#]&],{n,100}]
%o A378249 (PARI) f(p) = p++; while(!ispower(p), p++); p;
%o A378249 lista(nn) = apply(f, primes(nn)); \\ _Michel Marcus_, Dec 19 2024
%Y A378249 A version for prime powers (but starting with prime(k) + 1) is A345531.
%Y A378249 Positions of last appearances are A377283, complement A377436.
%Y A378249 Restriction of A377468 to the primes, for prime powers A000015.
%Y A378249 The opposite is A378035, restriction of A081676.
%Y A378249 The union is A378250.
%Y A378249 Run lengths are A378251.
%Y A378249 A000040 lists the primes, differences A001223.
%Y A378249 A000961 lists the powers of primes, differences A057820.
%Y A378249 A001597 lists the perfect powers, differences A053289, seconds A376559.
%Y A378249 A007916 lists numbers that are not perfect powers, differences A375706, seconds A376562.
%Y A378249 A069623 counts perfect powers <= n.
%Y A378249 A076411 counts perfect powers < n.
%Y A378249 A131605 lists perfect powers that are not prime powers.
%Y A378249 A377432 counts perfect powers between primes, zeros A377436, postpositives A377466.
%Y A378249 Cf. A007918, A023055, A031218, A045542, A052410, A065514, A076412, A188951, A216765, A377434.
%K A378249 nonn
%O A378249 1,1
%A A378249 _Gus Wiseman_, Nov 21 2024