A378035 Greatest perfect power < prime(n).

1, 1, 4, 4, 9, 9, 16, 16, 16, 27, 27, 36, 36, 36, 36, 49, 49, 49, 64, 64, 64, 64, 81, 81, 81, 100, 100, 100, 100, 100, 125, 128, 128, 128, 144, 144, 144, 144, 144, 169, 169, 169, 169, 169, 196, 196, 196, 216, 225, 225, 225, 225, 225, 243, 256, 256, 256, 256

Offset: 1

Author

Gus Wiseman, Nov 23 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 positive integer k at position prime(k).
-1-----4-------8-9------------16----------------25--27--------32------36----
===1=2===3===4=======5===6=======7===8=======9==========10==11==========12==
		

Crossrefs

Restriction of A081676 to the primes.

Positions of last appearances are also A377283.

A version for squarefree numbers is A378032.

The opposite is A378249 (run lengths A378251), restriction of A377468 to the primes.

The union is A378253.

Terms appearing exactly once are A378355.

Run lengths are A378356, first differences of A377283, complement A377436.

A000040 lists the primes, differences A001223.

A000961 lists the powers of primes, differences A057820.

A001597 lists the perfect powers, differences A053289.

A007916 lists the nonperfect powers, differences A375706.

A069623 counts perfect powers <= n.

A076411 counts perfect powers < n.

A080769 counts primes between perfect powers, prime powers A067871.

A131605 lists perfect powers that are not prime powers.

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

Cf. A000015, A007918, A031218, A045542, A052410, A065514, A076412, A188951, A216765, A345531, A377434, A378250.

Programs

Mathematica

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

PARI

a(n) = my(k=prime(n)-1); while (!(ispower(k) || (k==1)), k--); k; \\ Michel Marcus, Nov 25 2024

Python

from sympy import mobius, integer_nthroot, prime
def A378035(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(x-1+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    m = (p:=prime(n)-1)-f(p)
    return bisection(lambda x:f(x)+m,m,m) # Chai Wah Wu, Nov 25 2024