A378250 Perfect-powers x > 1 such that it is not possible to choose a prime y and a perfect-power z satisfying x > y > z.

4, 8, 16, 25, 32, 49, 64, 81, 100, 121, 128, 144, 169, 196, 216, 225, 243, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1600, 1681, 1728, 1764, 1849, 1936

Offset: 1

Author

Gus Wiseman, Nov 21 2024

Keywords

• nonn

Comments

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

Examples

			The first number line below shows the perfect-powers. The second shows the primes. The third is a(n).
-1-----4-------8-9------------16----------------25--27--------32------36----
===2=3===5===7======11==13======17==19======23==========29==31==========37==
       4       8              16                25            32
The terms together with their prime indices begin:
     4: {1,1}
     8: {1,1,1}
    16: {1,1,1,1}
    25: {3,3}
    32: {1,1,1,1,1}
    49: {4,4}
    64: {1,1,1,1,1,1}
    81: {2,2,2,2}
   100: {1,1,3,3}
   121: {5,5}
   128: {1,1,1,1,1,1,1}
   144: {1,1,1,1,2,2}
   169: {6,6}
   196: {1,1,4,4}
   216: {1,1,1,2,2,2}
   225: {2,2,3,3}
   243: {2,2,2,2,2}
   256: {1,1,1,1,1,1,1,1}
		

Crossrefs

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

The opposite is union of A378035, restriction of A081676.

Union of A378249, run-lengths are A378251.

A000040 lists the primes, differences A001223.

A000961 lists the powers of primes, differences A057820.

A001597 lists the perfect-powers, differences A053289, seconds A376559.

A007916 lists the non-perfect-powers, differences A375706, seconds A376562.

A069623 counts perfect-powers <= n.

A076411 counts perfect-powers < n.

A131605 lists perfect-powers that are not prime-powers.

A377432 counts perfect-powers between primes, zeros A377436, positive A377283, postpositive A377466.

Cf. A000015, A007918, A023055, A045542, A052410, A076412, A188951, A216765, A377431, A377434, A377468.

Programs

Mathematica

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Union[Table[NestWhile[#+1&,Prime[n],radQ[#]&],{n,100}]]