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A342871 a(n) = Sum_{k=1..n} floor(n^(1/k)), n >= 1.

%I A342871 #69 Apr 06 2021 12:04:36
%S A342871 1,3,5,8,10,12,14,17,20,22,24,26,28,30,32,36,38,40,42,44,46,48,50,52,
%T A342871 55,57,60,62,64,66,68,71,73,75,77,80,82,84,86,88,90,92,94,96,98,100,
%U A342871 102,104,107,109,111,113,115,117,119,121,123,125,127,129,131,133
%N A342871 a(n) = Sum_{k=1..n} floor(n^(1/k)), n >= 1.
%H A342871 Avid Rajai and David A. Corneth, <a href="/A342871/b342871.txt">Table of n, a(n) for n = 1..10000</a>
%F A342871 Lim_{n->infinity} a(n)/n = 2.
%F A342871 a(n) = 2*n + sqrt(n) + O(n^(1/3)).
%F A342871 Lim_{n->infinity} (a(n)/n - 2)*sqrt(n) = 1.
%F A342871 a(n) = A043000(n) + 1 for n >= 2.
%F A342871 a(n) = A255165(n) + n for n >= 2.
%F A342871 a(n) = A089361(n) + 2*n - 1 for n >= 2.
%F A342871 a(n) = n + Sum_{i=1..floor(log_2(n))} floor(n^(1/i) - 1).
%F A342871 If n is in A001597 then a(A001597(m)) - a(A001597(m)-1) = 2 + A253642(m), otherwise a(n) - a(n-1) = 2.
%F A342871 2 <= a(n)/n <= 9/4 iff n >= 4.
%F A342871 1 <= (a(n)/n - 2)*sqrt(n) <= 27/16 iff n >= 27.
%F A342871 2*n + sqrt(n) < a(n) <= 2*n + (27/16)*sqrt(n) iff n >= 27.
%t A342871 Table[Sum[Floor[n^(1/k)],{k,n}],{n,100}] (* _Giorgos Kalogeropoulos_, Mar 31 2021 *)
%o A342871 (PARI) a(n)=sum(k=1, n, sqrtnint(n,k)) \\ _Andrew Howroyd_, Mar 28 2021
%o A342871 (PARI) a(n) = if(n < 2, return(n)); my(c = logint(n, 2)); 2*n + sum(i = 2, c, sqrtnint(n, i)) - c \\ _David A. Corneth_, Mar 28 2021
%o A342871 (Python)
%o A342871 from sympy import integer_nthroot
%o A342871 def A342871(n):
%o A342871     c = 0
%o A342871     for k in range(1,n+1):
%o A342871         m = integer_nthroot(n,k)[0]
%o A342871         if m == 1:
%o A342871             return c+n-k+1
%o A342871         else:
%o A342871             c += m
%o A342871     return c # _Chai Wah Wu_, Apr 06 2021
%Y A342871 Cf. A043000, A255165, A089361, A001597, A253642.
%K A342871 nonn
%O A342871 1,2
%A A342871 _Avid Rajai_, Mar 28 2021