A378365 Next prime index after each perfect power, duplicates removed.

1, 3, 5, 7, 10, 12, 16, 19, 23, 26, 31, 32, 35, 40, 45, 48, 49, 54, 55, 62, 67, 69, 73, 79, 86, 93, 98, 100, 106, 115, 123, 130, 138, 147, 155, 163, 169, 173, 182, 192, 201, 211, 218, 220, 229, 241, 252, 264, 270, 275, 284, 296, 307, 310, 320, 328, 330, 343

Offset: 1

Author

Gus Wiseman, Nov 26 2024

Keywords

• nonn

Comments

Perfect powers (A001597) are 1 and numbers with a proper integer root, complement A007916.

Examples

			The first number line below shows the perfect powers. The second shows each n at position prime(n). To get a(n), we take the first prime between each pair of consecutive perfect powers, skipping the cases where there are none.
-1-----4-------8-9------------16----------------25--27--------32------36----
===1=2===3===4=======5===6=======7===8=======9==========10==11==========12==
		

Crossrefs

The opposite version is A377283.

Positions of first appearances in A378035.

First differences are A378251.

Union of A378356.

A000040 lists the primes, differences A001223.

A000961 lists the powers of primes, differences A057820.

A001597 lists the perfect powers, differences A053289.

A007916 lists the non perfect powers, differences A375706.

A069623 counts perfect powers <= n.

A076411 counts perfect powers < n.

A080769 counts primes between perfect powers.

A377432 counts perfect powers between primes, see A377434, A377436, A377466.

A378249 gives the least perfect power > prime(n), restriction of A377468.

Cf. A045542, A052410, A065514, A068361, A076412, A081676, A216765, A345531, A378250, A378253, A378355.

Programs

Mathematica

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Union[1+Table[PrimePi[n],{n,Select[Range[100],perpowQ]}]]

Formula

These are the distinct elements of the set {1 + A000720(A151800(n)), n>0}.