A096165 Prime powers with exponents that are themselves prime powers.

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227

Offset

1,1

Comments

A000040, A053810, A050376 and A082522 are subsequences;

a(n) = A000961(n+1) for 1<=n<=26.

Complement of A164345 with respect to A000961.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Formula

a(n) ~ n log n. - Charles R Greathouse IV, Oct 19 2015

Example

512=2^9=2^(3^2), A000961(118)=A000040(1)^A000961(118), therefore 512 is a term;
64=2^6, but 6 is not a prime power, therefore 64 is not a term.

Maple

F:= proc(t) local P;
P:= ifactors(t)[2];
nops(P) = 1 and (P[1][2]=1 or nops(numtheory:-factorset(P[1][2]))=1)
end proc:
select(F, [$2..1000]); # Robert Israel, Jul 20 2015

Mathematica

Select[Range@ 240, Or[PrimeQ@ #, PrimePowerQ@ # && PrimePowerQ@ FactorInteger[#][[1, 2]]] &] (* Michael De Vlieger, Jul 20 2015 *)

Prog

(Haskell)
a096165 n = a096165_list !! (n-1)
a096165_list = filter ((== 1) . a010055 . a001222) $ tail a000961_list
-- Reinhard Zumkeller, Nov 17 2011
(PARI) is(n)=while(1, if(!(n=isprimepower(n)), return(0), if(n==1, return(1)))) \\ Anders Hellström, Jul 19 2015
(PARI) ispp(n)=n==1 || isprimepower(n)
is(n)=ispp(isprimepower(n)) \\ Charles R Greathouse IV, Oct 19 2015
(Python)
from sympy import primepi, integer_nthroot, factorint
def A096165(n):
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x, k)[0]) for k in range(1, x.bit_length()) if len(factorint(k))<=1))
    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
    return bisection(f, n, n) # Chai Wah Wu, Sep 12 2024

Crossrefs

Cf. A000040, A000961, A010055, A001222, A050376, A053810, A082522, A164336, A164345.

Sequence in context: A000961 A128603 A195943 * A164336 A348263 A115919

Adjacent sequences: A096162 A096163 A096164 * A096166 A096167 A096168

Keyword

nonn,changed

Author

Reinhard Zumkeller, Jul 25 2004

Status

approved