A378249 - OEIS

%I #13 Dec 19 2024 10:25:06

%S 4,4,8,8,16,16,25,25,25,32,32,49,49,49,49,64,64,64,81,81,81,81,100,

%T 100,100,121,121,121,121,121,128,144,144,144,169,169,169,169,169,196,

%U 196,196,196,196,216,216,216,225,243,243,243,243,243,256,289,289,289

%N Least perfect power > prime(n).

%C Perfect-powers (A001597) are numbers with a proper integer root, complement A007916.

%C Which terms appear only once? Just 128, 225, 256, 64009, 1295044?

%e The first number line below shows the perfect powers. The second shows each prime.

%e -1-----4-------8-9------------16----------------25--27--------32------36------------------------49--

%e ===2=3===5===7======11==13======17==19======23==========29==31==========37======41==43======47======

%t radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;

%t Table[NestWhile[#+1&,Prime[n],radQ[#]&],{n,100}]

%o (PARI) f(p) = p++; while(!ispower(p), p++); p;

%o lista(nn) = apply(f, primes(nn)); \\ _Michel Marcus_, Dec 19 2024

%Y A version for prime powers (but starting with prime(k) + 1) is A345531.

%Y Positions of last appearances are A377283, complement A377436.

%Y Restriction of A377468 to the primes, for prime powers A000015.

%Y The opposite is A378035, restriction of A081676.

%Y The union is A378250.

%Y Run lengths are A378251.

%Y A000040 lists the primes, differences A001223.

%Y A000961 lists the powers of primes, differences A057820.

%Y A001597 lists the perfect powers, differences A053289, seconds A376559.

%Y A007916 lists numbers that are not perfect powers, differences A375706, seconds A376562.

%Y A069623 counts perfect powers <= n.

%Y A076411 counts perfect powers < n.

%Y A131605 lists perfect powers that are not prime powers.

%Y A377432 counts perfect powers between primes, zeros A377436, postpositives A377466.

%Y Cf. A007918, A023055, A031218, A045542, A052410, A065514, A076412, A188951, A216765, A377434.

%K nonn

%O 1,1

%A _Gus Wiseman_, Nov 21 2024