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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.
11
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
1,1
COMMENTS
Perfect-powers (A001597) are numbers with a proper integer root, complement A007916.
EXAMPLE
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}
MATHEMATICA
radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Union[Table[NestWhile[#+1&, Prime[n], radQ[#]&], {n, 100}]]
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.
Sequence in context: A368614 A330992 A246067 * A161226 A369566 A022560
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 21 2024
STATUS
approved