cp's OEIS Frontend

A075109 Odd perfect powers (1 together with numbers m^k, m odd, k >= 2).

%I A075109 #44 Feb 26 2025 01:55:16
%S A075109 1,9,25,27,49,81,121,125,169,225,243,289,343,361,441,529,625,729,841,
%T A075109 961,1089,1225,1331,1369,1521,1681,1849,2025,2187,2197,2209,2401,2601,
%U A075109 2809,3025,3125,3249,3375,3481,3721,3969,4225,4489,4761,4913,5041,5329,5625
%N A075109 Odd perfect powers (1 together with numbers m^k, m odd, k >= 2).
%H A075109 Reinhard Zumkeller, <a href="/A075109/b075109.txt">Table of n, a(n) for n = 1..10000</a>
%F A075109 Sum_{n>=1} 1/a(n) = 1 + Sum_{k>=2} mu(k)*(1-zeta(k)*(2^k-1)/2^k) = 1.2890375574... - _Amiram Eldar_, Dec 19 2020
%p A075109 q:= n-> n=1 or n::odd and igcd(seq(i[2], i=ifactors(n)[2]))>1:
%p A075109 select(q, [$1..6000])[];  # _Alois P. Heinz_, May 04 2022
%t A075109 Take[Union[Flatten[Table[a^b, {a, 1, 99, 2}, {b, 2, 15}]]], 50] (* _Alonso del Arte_, Nov 22 2011 *)
%o A075109 (Haskell)
%o A075109 a075109 n = a075109_list !! (n-1)
%o A075109 a075109_list = filter odd a001597_list  -- _Reinhard Zumkeller_, Oct 04 2012
%o A075109 (Magma) [1] cat [n : n in [3..6000 by 2] | IsPower(n) ]; // _Vincenzo Librandi_, Mar 31 2014
%o A075109 (PARI) isok(m) = (m==1) || ((m%2) && ispower(m)); \\ _Michel Marcus_, May 04 2022
%o A075109 (Python)
%o A075109 from sympy import mobius, integer_nthroot
%o A075109 def A075109(n):
%o A075109     def bisection(f,kmin=0,kmax=1):
%o A075109         while f(kmax) > kmax: kmax <<= 1
%o A075109         kmin = kmax >> 1
%o A075109         while kmax-kmin > 1:
%o A075109             kmid = kmax+kmin>>1
%o A075109             if f(kmid) <= kmid:
%o A075109                 kmax = kmid
%o A075109             else:
%o A075109                 kmin = kmid
%o A075109         return kmax
%o A075109     def f(x): return int(n-1+x+sum(mobius(k)*((integer_nthroot(x,k)[0]+1>>1)-1) for k in range(2,x.bit_length())))
%o A075109     return bisection(f,n,n) # _Chai Wah Wu_, Feb 25 2025
%Y A075109 Intersection of A001597 and A005408.
%Y A075109 Cf. A008683, A075090.
%K A075109 easy,nonn
%O A075109 1,2
%A A075109 _Zak Seidov_, Oct 11 2002
%E A075109 Definition clarified by _N. J. A. Sloane_, Dec 25 2021