A342871 - OEIS

%I #69 Apr 06 2021 12:04:36

%S 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 55,57,60,62,64,66,68,71,73,75,77,80,82,84,86,88,90,92,94,96,98,100,

%U 102,104,107,109,111,113,115,117,119,121,123,125,127,129,131,133

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

%H Avid Rajai and David A. Corneth, <a href="/A342871/b342871.txt">Table of n, a(n) for n = 1..10000</a>

%F Lim_{n->infinity} a(n)/n = 2.

%F a(n) = 2*n + sqrt(n) + O(n^(1/3)).

%F Lim_{n->infinity} (a(n)/n - 2)*sqrt(n) = 1.

%F a(n) = A043000(n) + 1 for n >= 2.

%F a(n) = A255165(n) + n for n >= 2.

%F a(n) = A089361(n) + 2*n - 1 for n >= 2.

%F a(n) = n + Sum_{i=1..floor(log_2(n))} floor(n^(1/i) - 1).

%F If n is in A001597 then a(A001597(m)) - a(A001597(m)-1) = 2 + A253642(m), otherwise a(n) - a(n-1) = 2.

%F 2 <= a(n)/n <= 9/4 iff n >= 4.

%F 1 <= (a(n)/n - 2)*sqrt(n) <= 27/16 iff n >= 27.

%F 2*n + sqrt(n) < a(n) <= 2*n + (27/16)*sqrt(n) iff n >= 27.

%t Table[Sum[Floor[n^(1/k)],{k,n}],{n,100}] (* _Giorgos Kalogeropoulos_, Mar 31 2021 *)

%o (PARI) a(n)=sum(k=1, n, sqrtnint(n,k)) \\ _Andrew Howroyd_, Mar 28 2021

%o (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 (Python)

%o from sympy import integer_nthroot

%o def A342871(n):

%o     c = 0

%o     for k in range(1,n+1):

%o         m = integer_nthroot(n,k)[0]

%o         if m == 1:

%o             return c+n-k+1

%o         else:

%o             c += m

%o     return c # _Chai Wah Wu_, Apr 06 2021

%Y Cf. A043000, A255165, A089361, A001597, A253642.

%K nonn

%O 1,2

%A _Avid Rajai_, Mar 28 2021