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A378035 Greatest perfect power < prime(n).

%I A378035 #22 Nov 26 2024 02:18:37
%S A378035 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,
%T A378035 100,100,100,100,100,125,128,128,128,144,144,144,144,144,169,169,169,
%U A378035 169,169,196,196,196,216,225,225,225,225,225,243,256,256,256,256
%N A378035 Greatest perfect power < prime(n).
%C A378035 Perfect powers (A001597) are 1 and numbers with a proper integer root, complement A007916.
%e A378035 The first number line below shows the perfect powers.
%e A378035 The second shows each positive integer k at position prime(k).
%e A378035 -1-----4-------8-9------------16----------------25--27--------32------36----
%e A378035 ===1=2===3===4=======5===6=======7===8=======9==========10==11==========12==
%t A378035 radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
%t A378035 Table[NestWhile[#-1&,Prime[n],radQ[#]&],{n,100}]
%o A378035 (PARI) a(n) = my(k=prime(n)-1); while (!(ispower(k) || (k==1)), k--); k; \\ _Michel Marcus_, Nov 25 2024
%o A378035 (Python)
%o A378035 from sympy import mobius, integer_nthroot, prime
%o A378035 def A378035(n):
%o A378035     def bisection(f,kmin=0,kmax=1):
%o A378035         while f(kmax) > kmax: kmax <<= 1
%o A378035         while kmax-kmin > 1:
%o A378035             kmid = kmax+kmin>>1
%o A378035             if f(kmid) <= kmid:
%o A378035                 kmax = kmid
%o A378035             else:
%o A378035                 kmin = kmid
%o A378035         return kmax
%o A378035     def f(x): return int(x-1+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
%o A378035     m = (p:=prime(n)-1)-f(p)
%o A378035     return bisection(lambda x:f(x)+m,m,m) # _Chai Wah Wu_, Nov 25 2024
%Y A378035 Restriction of A081676 to the primes.
%Y A378035 Positions of last appearances are also A377283.
%Y A378035 A version for squarefree numbers is A378032.
%Y A378035 The opposite is A378249 (run lengths A378251), restriction of A377468 to the primes.
%Y A378035 The union is A378253.
%Y A378035 Terms appearing exactly once are A378355.
%Y A378035 Run lengths are A378356, first differences of A377283, complement A377436.
%Y A378035 A000040 lists the primes, differences A001223.
%Y A378035 A000961 lists the powers of primes, differences A057820.
%Y A378035 A001597 lists the perfect powers, differences A053289.
%Y A378035 A007916 lists the nonperfect powers, differences A375706.
%Y A378035 A069623 counts perfect powers <= n.
%Y A378035 A076411 counts perfect powers < n.
%Y A378035 A080769 counts primes between perfect powers, prime powers A067871.
%Y A378035 A131605 lists perfect powers that are not prime powers.
%Y A378035 A377432 counts perfect powers between primes, zeros A377436, postpositives A377466.
%Y A378035 Cf. A000015, A007918, A031218, A045542, A052410, A065514, A076412, A188951, A216765, A345531, A377434, A378250.
%K A378035 nonn
%O A378035 1,3
%A A378035 _Gus Wiseman_, Nov 23 2024