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%I #21 Oct 20 2023 05:17:08
%S 1,1,1,5,5,5,5,13,22,22,22,22,22,22,22,38,38,38,38,38,38,38,38,38,63,
%T 63,90,90,90,90,90,122,122,122,122,158,158,158,158,158,158,158,158,
%U 158,158,158,158,158,207,207,207,207,207,207,207,207,207,207,207,207,207
%N Sum of perfect powers <= n.
%H Robert Israel, <a href="/A076407/b076407.txt">Table of n, a(n) for n = 1..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PerfectPower.html">Perfect Powers</a>.
%F a(n) = 1 - Sum_{k=2..floor(log_2(n))} mu(k) * (F(k, floor(n^(1/k))) - 1), where F(k, n) = Sum_{j=1..n} j^k = (Bernoulli(k+1, n+1) - Bernoulli(k+1, 1))/(k+1). - _Daniel Suteu_, Aug 19 2023
%e Sum of the 8 perfect powers <= 32: a(32) = 1+4+8+9+16+25+27+32 = 122.
%p N:= 100: # for a(1)..a(N)
%p V:= Vector(N,1):
%p pps:= {seq(seq(x^k,k=2..floor(log[x](N))),x=2..floor(sqrt(N)))}:
%p for y in pps do
%p V[y..N]:= V[y..N] +~ y
%p od:
%p convert(V,list); # _Robert Israel_, Oct 19 2023
%o (PARI)
%o F(k,n) = (subst(bernpol(k+1), x, n+1) - subst(bernpol(k+1), x, 1)) / (k+1);
%o a(n) = 1 - sum(k=2, logint(n,2), moebius(k) * (F(k, sqrtnint(n,k)) - 1)); \\ _Daniel Suteu_, Aug 19 2023
%Y Cf. A001597, A076408, A069623.
%K nonn
%O 1,4
%A _Reinhard Zumkeller_, Oct 09 2002
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